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my Maple cheat sheet


May 22, 2024   Compiled on May 22, 2024 at 3:44pm  [public]
1 OOP in Maple
1.1 How to use new object method calling in Maple 2021?
1.2 How to make a constructor for an Object?
1.3 How to make different constructors for an Object?
1.4 How to do OOP inheritance?
1.5 How to extend a base class and override its method with different one?
1.6 How to extend a class and call base class function from the extended class??
1.7 How to use object as user defined record inside a proc?
1.8 How to make copy of list of objects?
1.9 How to use OOP to implement ode solver?
1.10 How to make a complete OOP ode solver in Maple?
2 Differential equations
2.1 How to force dsolve to use specific method for solving?
2.2 How to find a particular solution to ODE?
2.3 How to find basis solutions for homogeneous ode?
2.4 How to solve a differential equation with initial conditions?
2.5 How to verify that the ODE solution given is correct?
2.6 How to know the type of ODE?
2.7 What packages to load for differential equations?
2.8 How to plot solution of differential equations?
2.9 On High precision. Using taylor to solve ODE
2.10 Obtain ODE in canonical coordinates (Lie symmetry)
2.11 How to use hint with symgen?
2.12 How to parse a single ode?
2.13 How to check if single ODE is valid?
2.14 How to check if an ode has \(y'\) in it?
2.15 How to check if an ode is linear ode?
2.16 How to find the order of an ode?
2.17 How to find the coefficients of a linear ode?
2.18 find order and degree of highest derivative
2.19 How to move all derivatives to one side in an equation?
2.20 How to obtains list of all derivatives in expression?
2.21 How to invert roles of dependent variable and independent variable in an ode?
2.22 How to find the indicial equation for an ODE?
2.23 How to write derivative
2.24 How to solve heat PDE in 1D in Maple 2017?
2.25 How to make Maple display diff(y(x),x) as y’(x) or as y’ ?
2.26 How to set boundary conditions for dsolve or pdsolve?
2.27 How force dsolve to use Lie?
2.28 How to do change of variables on the dependent variable for an ODE?
2.29 How to do change of variable on the independent variable for an ODE?
2.30 ODE change of variable on both dependent and independent variable?
2.31 How to make phase plot of first order ODE?
2.32 How to make phase plot of second order ODE?
2.33 How to move all terms with y to one side?
2.34 How to find all constants of integrations in an expression?
2.35 How to find if an ode is missing the dependent variable?
3 Symbolic manipulation, expression filtering, structured types
3.1 on maple data types
3.2 how to extract elements from a list based on some selection?
3.3 how to test if all elements of a matrix are integers?
3.4 How to select terms with sqrt or radical
3.5 How to find all csgn() and replace them by 1
3.6 How to find symbols inside csgn() in an expression?
3.7 How to replace all abs(expr) by expr
3.8 How to obtain list of all occurances of some function in an expression?
3.9 How to replace \(\ln (|x|)\) with \(\ln (x)\) in an expression?
3.10 How to find all signum functions in expression and simplify it?
3.11 How to replace all signum functions in expression by 1?
3.12 How to find if some function is present in an expression
3.13 How to find all functions in an expression?
3.14 find all functions except builtin math functions
3.15 How to obtain a list of all arguments of function?
3.16 Find functions whose first argument is \(z\)
3.17 Find functions whose second argument is \(t\)?
3.18 How to use select with own type to find subexpressions?
3.19 How to write structured types to match some expressions?
3.20 select only indexed variables from an expression
3.21 Given an expression, how to find all variables and functions in it?
3.22 How to check if an expression is integer, when it has symbols in it?
3.23 How to find parameters such as \(\pi \) in an expression?
3.24 How to find all derivatives \(y'(x)\) in an expression?
3.25 How combine log terms?
3.26 Find position in a list of items that are not numeric
3.27 How to change \(\arctan (y,x)\) to \(\arctan \left (\genfrac {}{}{}{}{y}{x}\right )\) in an expression
3.28 How to find all symbols that represent variables in an expression?
3.29 How to change first argument of function?
3.30 How to change last argument of function?
3.31 How to remove last argument of function?
3.32 How to find a pattern inside an expression
3.33 How to find parts of a Sum?
3.34 find if sequence or list is inside another and the indecies
3.35 find if some type inside some expression and its location
3.36 How to replace generic function inside derivative?
3.37 Examples how to match types
3.38 On the order of terms when using indents
4 How to convert Mathematica expression to Maple?
5 How to debug internal procedures, such as dsolve?
6 How to display source code of a function?
7 How to display trace of a function as it runs in maple?
8 How to display a build in function code?
9 How to build a LIST or a SET on the fly?
10 make function display more information
11 How to plot a function?
12 How to run maple from command line?
13 How to use matrices in maple?
14 return more than value from a procedure
15 How does maple handle procedure arguments?
16 How to define your own data types?
17 find max element and position in matrix
18 How to create a package?
19 How to convert from floating point to Hex?
20 How to find taylor series expansion of functions?
21 How to print elements of a matrix?
22 How to find determinant of matrix?
23 How to generate Hilber matrix?
24 How to plot matrix data?
25 How to catch an error from a proc()?
26 How to convert 3456 to 3,456 ?
27 How to use units ?
28 How to evaluate catlan number and other sums?
29 write a text file that contains a package, and load it
30 How to find what packages are included in maple
31 How to plot the gradiant vector field?
32 How to put the digits of Pi into a list?
33 Digits of PI in maple and mma
34 How to find where functions are?
35 how to use laplace transform?
36 questions I have
37 3D plotting
38 How to raise each element in a list to a power?
39 How to generate a sequence with any increment?
40 What shortcuts are there for matrix manipulation?
41 How to solve a set of equations for the derivative?
42 How to solve a set of equations for differentials?
43 How to plot binary tree
44 Problem 12.4 chapter 4, Boas book
45 example of doing convergence test in maple
46 Problem ch 14, 3.18, Boas book. contour integration
47 How to find multiple roots to an equation such as \(sin(x) = 0\)
48 Dr Basti Associated Legendre
49 Understanding conformal mapping in maple
50 Hide tilda character when using assumption
51 Fourier series in maple
52 How to plot graphs next to each others in a grid like fashion
53 How to generate Pi on X-axis
54 How to make output from FunctionAdvisor look better?
55 How to do partial fractions?
56 How to generate sequence sum symbolically
57 Nice plot from Maple
58 How to check if 2 expressions are the same?
59 converting series to factorials
60 How to find what new additions made to Maple?
61 Maple can’t solve laplace equation and numerically
62 Some Maple Matrix operations
63 How set diagonal elements to some value, say 1?
64 How to multiply roots of a polynomial?
65 How to plot a surface in 3D?
66 How to convert trigs to sinc function in an expression
67 How to find NullSpace and ColumnSpace of a matrix?
68 How to fix the interface to using Maple notation for input?
69 How to find all solutions using allvalues ?
70 Adding only to diagonal of a matrix
71 How to search help for updates on some package
72 How to work with groups in worksheet
73 How to read code into worksheet?
74 Code editors for Maple
75 How to find if package is module or table?
76 How to replace a string?
77 How to use geometry and plottools ?
78 How to simplify log expressions ?
79 How to simplify hyperbolic expression ?
80 How to create text file and append string to it?
81 How to search packages and libraries?
82 How to numerically solve a BVP ode and plot the solution?
83 How to display on screen for specific width?
84 Maple IDE links
85 loading, remove and finding what packages loaded
86 some rules of thumbs when using Maple
87 How to make multiple assumptions on a symbol?
88 How to check if expression is an equation?
89 How to check if expression is a set?
90 How to export a plot to PDF?
91 How to find all roots of complex number
92 How to convert matrix of matrices to a matrix?
93 How to do pattern matching in Maple?
93.1 Example 1
93.2 Example 2
94 How to find trig indetities?
95 How to find directional derivative of scalar function?
96 How to check if name is assigned a value?
97 How to simplify \(e^{\ln (x)+\ln (y)}\)
98 Basis for Null space, Row space and column space of matrix
99 How to do Gaussian elimination on a Matrix?
100 How to find Reduced Echelon form of a Matrix?
101 How add a new row to bottom of matrix?
102 How to find the cofactor matrix of a matrix?
103 How to normalize eigenvectors?
104 How to typeset \(\hslash \)?
105 How to find the Curl of a vector?
106 See all steps in RREF of an augmented matrix
107 How to find column space of matrix?
108 Show step by step. Calculus problem and differential equations.
109 How to obtain list of files with some extension in folder?
110 How to delete lines from text file that contains some string?
111 How to truncate a polynomial?
112 How to make a local declare like block inside a proc?
113 Using short name for a proc inside nested modules intead of long name
114 Remove duplicates objects in a list based on condition on a field
115 How to remove duplicates Vectors from a list?
116 How to find all poles and their order of a rational function?
117 find series of function with specific number of terms
118 How to call sibling’s proc without making the sibling module exported?
119 Convert time to use seconds instead of milliseconds
120 Change the summation index letter

1 OOP in Maple

1.1 How to use new object method calling in Maple 2021?
1.2 How to make a constructor for an Object?
1.3 How to make different constructors for an Object?
1.4 How to do OOP inheritance?
1.5 How to extend a base class and override its method with different one?
1.6 How to extend a class and call base class function from the extended class??
1.7 How to use object as user defined record inside a proc?
1.8 How to make copy of list of objects?
1.9 How to use OOP to implement ode solver?
1.10 How to make a complete OOP ode solver in Maple?

1.1 How to use new object method calling in Maple 2021?

In Maple 2021, it is now possible to use object:-method(arg) notation. This makes is easier to use OOP in maple. To do this, use _self as follows

restart; 
 
module person() 
 option object; 
 local name::string:=""; 
 local age::integer:=0; 
 
 export get_name::static:=proc(_self,$) 
   return _self:-name; 
 end proc; 
 
 export set_name::static:=proc(_self,name::string,$) 
   _self:-name:=name; 
 end proc; 
 
 export get_age::static:=proc(_self,$) 
   return _self:-age; 
 end proc; 
 
 export set_age::static:=proc(_self,age::integer,$) 
   _self:-age:=age; 
 end proc; 
 
end module;
 

And now make an object and use it as follows

o:=Object(person) 
                  o := Object<<1846759887808>> 
 
o:-get_name(); 
             "" 
 
o:-get_age(); 
             0 
 
o:-set_name("joe doe"); 
o:-get_name(); 
           "joe doe"
 

1.2 How to make a constructor for an Object?

Add ModuleCopy proc in the class. This will automatically be called to initialize the object.

Here is an example

restart; 
 
module ODE() 
 option object; 
 local ode:=NULL; 
 local y::symbol; 
 local x::symbol; 
 local sol; 
 
 export ModuleCopy::static := proc( _self::ODE, proto::ODE, ode, func, $ ) 
    print("Initilizing object with with args: ", [args]); 
    _self:-ode:= ode; 
    _self:-y:=op(0,func); 
    _self:-x:=op(1,func); 
 end proc; 
 
 export dsolve::static:=proc(_self,$) 
   _self:-sol := :-dsolve(ode,y(x)); 
 end proc; 
 
 export get_sol::static:=proc(_self,$) 
   return sol; 
 end proc; 
 
end module;
 

And now make an object and use it as follows

o:=Object(ODE, diff(y(x),x)+y(x)=sin(x), y(x)); 
o:-dsolve(): 
o:-get_sol(); 
 
        #y(x) = -1/2*cos(x) + 1/2*sin(x) + exp(-x)*_C1
 

So a constructor just makes it easier to initialize the object without having to make a number of set() calls to initialize each memeber data.

1.3 How to make different constructors for an Object?

This is done using overload with different ModuleCopy proc in the class.

Here is an example. Lets make a constructor that takes an ode and initial conditions, and one that only takes an ode with no initial conditions.

restart; 
 
module ODE() 
 option object; 
 local ode:=NULL; 
 local y::symbol; 
 local x::symbol; 
 local ic:=NULL; 
 local sol; 
 
 export ModuleCopy::static:= overload( 
    [ 
       proc( _self::ODE, proto::ODE, ode, func, $ ) option overload; 
          _self:-ode:= ode; 
          _self:-y:=op(0,func); 
          _self:-x:=op(1,func); 
       end proc, 
 
       proc( _self::ODE, proto::ODE, ode, func, ic, $ ) option overload; 
          _self:-ode:= ode; 
          _self:-y:=op(0,func); 
          _self:-x:=op(1,func); 
          _self:-ic :=ic; 
       end proc 
    ] 
 ); 
 
 export dsolve::static:=proc(_self,$) 
    if evalb(ic=NULL) then 
       sol := :-dsolve(ode,y(x)); 
    else 
       sol := :-dsolve([ode,ic],y(x)); 
    fi; 
 end proc; 
 
 export get_sol::static:=proc(_self,$) 
   return sol; 
 end proc; 
 
end module;
 

And now use it as follows

o:=Object(ODE, diff(y(x),x)+y(x)=sin(x), y(x), y(0)=0); 
o:-dsolve(): 
o:-get_sol(); 
 
         #y(x) = -1/2*cos(x) + 1/2*sin(x) + 1/2*exp(-x) 
 
o:=Object(ODE, diff(y(x),x)+y(x)=sin(x), y(x)); 
o:-dsolve(): 
o:-get_sol(); 
 
    #y(x) = -1/2*cos(x) + 1/2*sin(x) + exp(-x)*_C1
 

1.4 How to do OOP inheritance?

In the child class you want to extend from the parent class, add option object(ParentName);

Here is an example

restart; 
module ODE() 
   option object; 
   local ode; 
 
   export set_ode::static:=proc(_self,ode,$) 
     _self:-ode :=ode; 
   end proc; 
 
  export get_ode::static:=proc(_self,$) 
     return _self:-ode; 
   end proc; 
end module; 
 
#create class/module which extends the above 
module second_order_ode() 
   option object(ODE); 
   export get_ode_order::static:=proc(_self,$) 
      return 2; 
   end proc; 
end module;
 

In the above second_order_ode inherts all local variables and functions in the ODE class and adds new proc. Use as follows

o:=Object(second_order_ode); #create an object  instance 
o:-set_ode(diff(y(x),x)=sin(x)); 
o:-get_ode(); 
o:-get_ode_order();
 

Note that the child class can not have its own variable with the same name as the parent class. This is limitation. in C++ for example, local variables in extended class overrides the same named variable in the parent class.

Even if the variable have different type, Maple will not allow overriding. For example, this will fail

restart; 
module ODE() 
   option object; 
   local ode; 
   local id::integer; 
 
   export set_ode::static:=proc(_self,ode,$) 
     print("Enter ode::set_ode"); 
     _self:-ode :=ode; 
   end proc; 
 
  export get_ode::static:=proc(_self,$) 
     return _self:-ode; 
   end proc; 
end module; 
 
module second_order_ode() 
   option object(ODE); 
   local id::string; 
   export get_ode_order::static:=proc(_self,$) 
      return 2; 
   end proc; 
end module; 
 
    Error, (in second_order_ode) local `id` is declared more than once
 

There might be a way to handle this, i.e. to somehow exlicitly tell Maple to override parant class proc or variable name in the child. I do not know now. The above is using Maple 2021.1

1.5 How to extend a base class and override its method with different one?

This is called polymorphism in OOP. This is a base class animal_class which has make_sound method. This method acts just as a place holder (interface), which the extending class must extends (override) with an actual implementation.

The class is extended to make cat class and implementation is made.

module animal_class() 
 option object; 
 
 export make_sound::static:=proc(_self,$) 
     error("Not implemented, must be overriden"); 
 end proc; 
end module; 
 
%--------------- 
module cat_class() 
 option object(animal_class); 
 
 make_sound::static:=proc(_self,$)  #note. DO NOT USE export 
     print("mewooo"); 
 end proc; 
 
end module;
 

And now

o:=Object(animal_class); 
o:-make_sound(); 
 
Error, (in anonymous procedure) Not implemented, must be overriden
 

The above is by design. As the animal_class is meant to be extended to be usable.

my_cat:=Object(cat_class); 
my_cat:-make_sound(); 
 
                            "mewooo"
 

So the base class can have number of methods, which are all meant to be be have its implementation provided by an extending class. Each class which extends the base class must provide implementation.

1.6 How to extend a class and call base class function from the extended class??

Once a base class is extended, all methods in the base class become part of the extending class. So to call a base class method just use same way as if calling any other method in the extending class itself.

Here is an example.

module person() 
    option object; 
    local age::integer:=100; 
 
    export ModuleCopy::static:= proc(_self,proto::person, age::integer,$) 
           _self:-age := age; 
    end proc; 
 
    local base_class_method::static:=proc(_self,$) 
        print("In base class method..."); 
    end proc; 
 
 
end module; 
 
#---- extend the above class 
module young_person() 
   option object(person); 
   export process::static:=proc(_self,$) 
      print("In young_person process"); 
     _self:-base_class_method(); 
   end proc; 
end module;
 

Here is an example of usage

o:=Object(young_person,20); 
o:-process() 
 
                   "In young_person process" 
                   "In base class method..."
                                                                                    
                                                                                    
 

The above in Maple 2023.1

1.7 How to use object as user defined record inside a proc?

A Maple Object can be used a record type in other languags, such as Ada or Pascal. This example shows how to define a local type inside a proc and use it as record.

restart; 
foo:=proc(the_name::string,the_age::integer)::person_type; 
     local module person_type() #this acts as a record type 
         option object; 
         export the_name::string; 
         export the_age::integer; 
     end module; 
 
     local person::person_type:=Object(person_type); 
 
     person:-the_name:=the_name; 
     person:-the_age:=the_age; 
     return person; 
 end proc; 
 
o:=foo("joe doe",100); 
 
o:-the_name; 
                           "joe doe" 
o:-the_age; 
                              100
 

In the above person is local variable of type person_type. In the above example, the local variable was returned back to user. But this is just an example. One can declare such variables and just use them internally inside the proc only. This method helps one organize related variables into one record/struct like type. The type can also be made global if needed.

1.8 How to make copy of list of objects?

Suppose we have list L1 of objects and we want to copy this list to another list, say L2. If we just do L2 = L1 then this will not make an actual copy as any changes to L1 are still reflected in L2. The above only copies the reference to the same list.

To make a physical copy, need to use the copy command as follows

restart; 
 
module person_type() 
  option object; 
  export the_name::string:="doe joe"; 
  export the_age::integer:=100; 
end module; 
 
L1:=[]; 
for N from 1 to 5 do 
    o:=Object(person_type); 
    o:-the_name:=convert(N,string); 
    L1:= [ op(L1), o]; 
od: 
 
L2:=map(Z->copy(Z),L1):
 

Now making any changes to L1 will not affect L2. If we just did L2 = L1 then both will share same content which is not what we wanted.

1.9 How to use OOP to implement ode solver?

This is a basic example of using OOP in Maple to implement an ode solver. There is a base module called ode_base_class (I will be using class instead of module, as this is more common in OOP)

This program will for now support first order and second order ode’s only.

The base ode class will contain the basic operations and data which is common to both first order and second order ode’s.

Next, we will make a first order ode class, and second order ode class. Both of these will extend the base ode class.

Next, we will have more ode classes. For example, for first order ode, there will be linear first order ode class, and separable first order ode class, and Bernoulli ode class and so on. Each one these classess will extend the first order ode class which in turn extends the base ode class.

Same for second order ode’s. There will be second order constant coefficients ode class, and second order variable coefficient ode class and so on. Each one of these classes will extend the second order ode class which in turn extends the base ode class.

Let base_ode_class be the base of an ode class which will contain all the neccessary methods that are generic and applicable to an ode of any order and type.

These can be the ode expression itself, its order and any other data and operations which applicable to any ode type.

Now we want to create a first order class. This will have its own operations and private variables that are specific and make sense only to any first order ode.

Then these will be the first order separable ode class, which has methods that implement solving the ode using separable method and has other methods which makes sense only for first order separable ode. The following diagram is partial illustration of the is-A relation among possible classes.

First we define the base ode class and define private variables and method that are common to any ode type.

restart; 
 
module base_ode_class() 
  option object; 
  local the_ode; 
  local the_order::integer; 
 
  export get_ode::static:=proc(_self,$) 
         RETURN(_self:-the_ode); 
  end proc; 
 
  export get_order::static:=proc(_self,$) 
         RETURN(_self:-the_order); 
  end proc; 
 
end module;
 

Note that the base ode class does not have constructor. Since it is meant to be extended only.

The following is the first order ode class.

module first_order_ode_class() 
  option object(base_ode_class); 
 
  local  initial_conditions; 
  local  solution_found; 
 
  export get_IC::static:=proc(_self,$) 
         RETURN(_self:-initial_conditions); 
  end proc; 
 
  export get_solution::static:=proc(_self,$) 
         RETURN(_self:-solution_found); 
  end proc; 
 
  export verify_solution::static:=proc(_self,$)::truefalse; 
      #code to verify if the solution found is valid or not 
      #using odetest() 
  end proc; 
 
end module;
 

The following is the first order separable ode class which extends the above first order ode class.

module first_order_separable_ode_class() 
  option object(first_order_ode_class); 
  local f,g;  #ode of form  y'=f(x)*g(y) 
 
  export ModuleCopy::static:= proc(_self,proto::first_order_separable_ode_class, 
                                   ode, IC,$) 
      _self:-the_ode  := ode; 
      _self:-initial_conditions := IC; 
  end proc; 
 
  export dsolve::static:=proc(_self,$) 
         #solve the ode for now we will use Maple but in my code 
         #I have my own solver ofcourse. 
         _self:-solution_found:=  :-dsolve([_self:-the_ode, _self:-initial_conditions]); 
  end proc; 
end module;
 

In the above, when we create a instance of first_order_separable_ode_class then it now have the whole chain of classes into one. i.e. first order separable class extending the first order class which in turn extends the base ode class. For example

o:=Object(first_order_separable_ode_class,diff(y(x),x)=3*sin(x)*y(x),y(0)=1) 
o:-dsolve(); 
o:-get_solution() 
       #y(x) = exp(3)*exp(-3*cos(x)) 
o:-get_ode() 
       #diff(y(x), x) = 3*sin(x)*y(x)
 

The above calls will all work, even though the first order separable class has no method in it called get_ode but it extends a class which does, hence it works as it.

Now we will do the same for second order ode’s.

Advantage of this design, is that methods in base classes that are being extended can be reused by any class which extends them. Only methods that applies and specific to the lower level classes need to be implemented.

As we add more specific solvers, we just have to extend the base classes and the new solvers just need to implement its own specific dsolve and any specific methods and data that it needs itself.

Ofcourse in practice the above design is not complete as is. The user should not have to specify which class to instantiate, as user does not care what the ode type or class it is. They just want to to do

o:=Object(ode_class,diff(y(x),x)=3*sin(x)*y(x),y(0)=1) 
o:-dsolve(); 
o:-get_solution()
 

To solve this problem we have to make a factory method which is called to make the correct instance of the class and return that to the user. The factory method figures out the type of ode and it creates the correct instance of the correct class and returns that. So the call will become

o := make_ode_object( diff(y(x),x)=3*sin(x)*y(x), y(0)=1) 
o:-dsolve(); 
o:-get_solution()
 

The function make_ode_object above is the main interface the user will call to make an ode object.

This will be explained next with examples. One possibility is to make the factory function a global function or better, a public method in a utility module. For now, it is given here as stadalone function for illustration. The user calls this method to make an object of the correct instance of the ode. Here is complete implementation of all the above including the factory method.

#factory method. Makes objects for users 
make_ode_object:=proc(ode::`=`,func::function(name)) 
   local x,y,the_order; 
   y:=op(0,func); 
   x:=op(1,func); 
   the_order := PDEtools:-difforder(ode,x); 
   if the_order=1 then 
        RETURN(first_order_ode_class:-make_ode_object(ode,func)); 
   elif the_order=2 then 
        #RETURN(second_order_ode_class:-make_ode_object(ode,func)); 
        #implement later 
        NULL; 
   else 
        error "Only first and second order ode's are currently supported"; 
   fi; 
 
end proc: 
 
######################### 
module base_ode_class() 
  option object; 
  local the_ode; 
  local the_order::integer; 
 
  #methods bound to the object 
  export get_ode::static:=proc(_self,$) 
         RETURN(_self:-the_ode); 
  end proc; 
 
  export get_order::static:=proc(_self,$) 
         RETURN(_self:-the_order); 
  end proc; 
 
end module: 
################## 
module first_order_ode_class() 
  option object(base_ode_class); 
 
  local  initial_conditions; 
  local  solution_found; 
 
  #public factory method not bound to the object. 
  export make_ode_object:=proc(ode::`=`,func::function(name)) 
     local x,y,ode_type::string; 
     y:=op(0,func); 
     x:=op(1,func); 
     ode_type:="separable"; #code here which determined first order ode type 
     if ode_type="separable" then 
        RETURN( Object(first_order_separable_ode_class,ode,func)); 
     elif ode_type="linear" then 
         RETURN( Object(first_order_linear_ode_class,ode,func)); 
     fi; 
     #more ode types added here 
  end proc; 
 
  #methods bound to the object 
  export get_IC::static:=proc(_self,$) 
         RETURN(_self:-initial_conditions); 
  end proc; 
 
  export get_solution::static:=proc(_self,$) 
         RETURN(_self:-solution_found); 
  end proc; 
 
  export verify_solution::static:=proc(_self,$)::truefalse; 
      #code to verify if the solution found is valid or not 
      #using odetest() 
  end proc; 
 
end module: 
 
################## 
module first_order_separable_ode_class() 
  option object(first_order_ode_class); 
  local f,g;  #ode of form  y'=f(x)*g(y) 
 
  export ModuleCopy::static:= proc(_self,proto::first_order_separable_ode_class,ode,func::function(name),$) 
      _self:-the_ode  := ode; 
  end proc; 
 
  export dsolve::static:=proc(_self,$) 
         #print("Enter first_order_separable_ode_class:-dsolve"); 
         #solve the ode for now we will use Maple but in my code 
         #I have my own solver ofcourse. 
         _self:-solution_found:=  :-dsolve(_self:-the_ode); 
         NULL; 
  end proc; 
end module:
 

It is used as follows

o:=make_ode_object(diff(y(x),x)=sin(x)*y(x),y(x)); 
o:-dsolve(); 
o:-get_solution(); 
                    y(x) = c__1 exp(-cos(x))
 

1.10 How to make a complete OOP ode solver in Maple?

Here, I will start making a complete small OOP ode solver in Maple.

At each step more classes are added and enhanced until we get a fully working small ode solver based on OOP design that solves a first and second order ode, this is to show how it all works. Another solvers can be added later by simply extending the base class.

The base class is called Base_ode_class. There will be Second_order_ode_class and First_order_ode_class and these classes extend the base ode class. We can later add higher_order_ode_class.

Next, there are different classes which extend these. There is First_order_linear_ode_class and First_order_separable_ode_class and so on, and these extend the First_order_ode_class.

For example, if a user wanted to solve a first order ode which happend to be say separable, then object of class First_order_separable_ode_class will be created and used.

Since the user does not know and should not know what object to create, then the factory class will be used. The factory class is what the user initially calls to make the ode object.

It is the factory class which determines which type of class to instantiate based on the ode given.

The factory class is singleton (standard module in Maple, not of type object), which has the make_ode method which is called by the user. This method parses the ode and determines its order and then based on the order determine which subclass to use, and then instantiate this and returns the correct object to the user to use.

This object will have the dsolve method and other methods the user can use on the object.

The make_ode method in the factory module accepts only the ode itself the function such as \(y(x)\). A typical use is given below

ode :=  ODE_factoy_class:-make_ode( diff(y(x),x)=sin(x), y(x) ); 
ode:-set_IC(....); 
ode:-set_hint("the hint"); 
..... 
 
ode:-dsolve();  #solves the ode 
ode:-is_solved(); #checks if ode was successfully solved 
ode:-verify_sol(); #verifies the solution using maple odetest() 
ode:-is_sol_verified(); #asks if solution is verified 
ode:-get_number_of_sol(); #returns number of solutions, 1 or 2 etc... 
ode:-get_sol(); #returns the solutions found in list 
#and more method 
...
 

Examples at the end will show how all the above works on actual odes’s.

The initial call to make an ode does not have initial conditions, or hint and any other parameters. This is so to keep the call simple. As the factory method only makes the concrete ode object.

Additional methods are then used to add more information if needed by using the returned object itself, such as initial conditions, and hint and so on before calling the dsolve method on the object.

Here is a very basic setup which include the base ode class and extended to first order two subclasses for now.

restart; 
ODE_factory_class :=module() 
   #notice, normal module. No option object. 
 
   export make_ode:=proc(ode::`=`,func::function(name),$) 
       local dep_variables_found::list,item; 
       local y::symbol; 
       local x::symbol; 
       local ode_order::integer; 
 
       if nops(func)<>1 then 
           error("Parsing error, dependent variable must contain one argument, found ", func); 
       fi; 
       y:=op(0,func); 
       x:=op(1,func); 
       if not has(ode,y) then 
          error ("Supplied ode ",ode," has no ",y); 
       fi; 
 
       if not has(ode,x) then 
          error ("Supplied ode ",ode," has no ",x); 
       fi; 
 
       if not has(ode,func) then 
           error ("Supplied ode ",ode," has no ",func); 
       fi; 
 
       ode_order := PDEtools:-difforder(ode,x); 
 
       #this will check that the dependent variable will show with 
       #SAME argument in the ode. i.e. if y(x) and y(t) show up in same ode, it 
       #will throw exception, which is what we want. 
 
       try 
           dep_variables_found := PDEtools:-Library:-GetDepVars([y],ode); 
       catch: 
           error lastexception; 
       end try; 
 
       #now go over dep_variables_found and check the independent 
       #variable is same as x i.e. ode can be y'(z)+y(z)=0 but function is y(x). 
       for item in dep_variables_found do 
           if not type(item,function) then 
              error("Parsing error. Expected ",func," found ",item," in ode"); 
           else 
              if op(1,item) <> x then 
                  error("Parsing error. Expected ",func," found ",item," in ode"); 
              fi; 
           fi; 
       od; 
 
      #now go over all indents in ode and check that y only shows as y(x) and not as just y 
      #as the PDEtools:-Library:-GetDepVars([_self:-y],ode) code above does not detect this. 
      #i.e. it does not check  y'(x)+y=0 
 
      if numelems(indets(ode,identical(y))) > 0 then 
           error("Parsing error, Can not have ",y," with no argument inside ",ode); 
      fi; 
 
      if ode_order=1 then 
         RETURN(make_first_order_ode(ode,y,x)); 
      elif ode_order=2 then 
         RETURN(make_second_order_ode(ode,y,x)); 
      else 
         RETURN(make_higher_order_ode(ode,y,x)); 
      fi; 
  end proc; 
 
  local make_first_order_ode:=proc(ode::`=`,y::symbol,x::symbol) 
     #decide on what specific type the ode is, and make instant of it 
 
     if first_order_ode_quadrature_class:-is_quadrature(ode,y,x) then 
        RETURN(Object(first_order_ode_quadrature_class,ode,y,x)); 
     elif first_order_ode_linear_class:-is_linear(ode,y,x) then 
        RETURN(Object(first_order_ode_linear_class,ode,y,x)); 
     fi; #and so on 
  end proc; 
 
  local make_second_order_ode:=proc(ode::`=`,y::symbol,x::symbol) 
     #decide on what specific type the ode is, and make instant of it 
     #same as for first order 
  end proc; 
end module; 
#------------------------- 
module solution_class() 
   option object; 
   local the_solution; 
   local is_verified_solution::truefalse:=false; 
   local is_implicit_solution::truefalse:=false; 
 
   export ModuleCopy::static:= proc(_self,proto::solution_class,the_solution::`=`,is_implicit_solution::truefalse,$) 
       _self:-the_solution:=the_solution; 
       _self:-is_implicit_solution:=is_implicit_solution; 
   end proc; 
 
   export get_solution::static:=proc(_self,$) 
        RETURN(_self:-the_solution); 
   end proc; 
 
   export is_verified::static:=proc(_self,$) 
        RETURN(_self:-is_verified_solution); 
   end proc; 
 
   export is_implicit::static:=proc(_self,$) 
        RETURN(_self:-is_implicit_solution); 
   end proc; 
 
   export is_explicit::static:=proc(_self,$) 
        RETURN(not(_self:-is_implicit_solution)); 
   end proc; 
 
   export verify_solution::static:= overload( 
   [ 
      proc(_self, ode::`=`,$) option overload; 
          local stat; 
          stat:= odetest(_self:-the_solution,ode); 
          if stat=0 then 
             _self:-is_verified_solution:=true; 
          else 
            if simplify(stat)=0 then 
                _self:-is_verified_solution:=true; 
            else 
                _self:-is_verified_solution:=false; 
            fi; 
         fi; 
      end, 
 
      proc(_self, ode::`=`,IC::list,$) option overload; 
          local stat; 
          stat:= odetest([_self:-the_solution,IC],ode); 
          if stat=[0,0] then 
             _self:-is_verified_solution:=true; 
          else 
            if simplify(stat)=[0,0] then 
                _self:-is_verified_solution:=true; 
            else 
                _self:-is_verified_solution:=false; 
            fi; 
         fi; 
      end 
   ]); 
end module: 
#------------------------- 
 
module ODE_base_class() 
   option object; 
   local y::symbol; 
   local x::symbol; 
   local func::function(name); #y(x) 
   local ode::`=`; 
   local ode_order::posint; 
   local IC::list:=[]; 
   local parsed_IC::list:=[]; 
   local the_hint::string:=""; 
   local solutions_found::list(solution_class):=[]; 
 
  #exported getters methods 
  export get_ode::static:=proc(_self,$) 
         RETURN(_self:-ode); 
  end proc; 
 
  export get_x::static:=proc(_self,$) 
         RETURN(_self:-x); 
  end proc; 
 
  export get_y::static:=proc(_self,$) 
         RETURN(_self:-y); 
  end proc; 
 
  export get_ode_order::static:=proc(_self,$) 
         RETURN(_self:-ode_order); 
  end proc; 
 
  export get_IC::static:=proc(_self,$) 
         RETURN(_self:-IC); 
  end proc; 
 
  export get_parsed_IC::static:=proc(_self,$) 
         RETURN(_self:-parsed_IC); 
  end proc; 
 
  export get_sol::static:=proc(_self,$) 
         local L:=Array(1..0): 
         local sol; 
         for sol in _self:-solutions_found do 
             L ,= sol:-get_solution(); 
         od; 
         RETURN(convert(L,list)); 
  end proc; 
 
  #exported setters methods 
  export set_hint::static:=proc(_self,hint::string,$) 
         #add code to check if hint is valid 
         _self:-the_hint:=hint; 
  end proc; 
end module; 
 
#------------------------- 
module first_order_ode_quadrature_class() 
  option object(ODE_base_class); 
  local f,g;  #ode of form  y'=f(x)*g(y) 
 
  #this method is not an object method. It is part of the module but does 
  #not have _self. It is called by the factory class to find if the ode 
  #is of this type first 
  export is_quadrature:=proc(ode::`=`,y::symbol,x::symbol)::truefalse; 
         RETURN(true); #for now 
  end proc; 
 
  export ModuleCopy::static:= proc(_self,proto::first_order_ode_quadrature_class,ode::`=`,y::symbol,x::symbol,$) 
      _self:-ode  := ode; 
      _self:-y  := y; 
      _self:-x  := x; 
      _self:-func  := _self:-y(_self:-x); 
      _self:-ode_order :=1; 
  end proc; 
 
  export dsolve::static:=proc(_self,$) 
         local sol,o; 
         #print("Enter first_order_ode_quadrature_class:-dsolve"); 
         #solve the ode for now we will use Maple but in my code 
         #I have my own solver ofcourse. 
         sol:= :-dsolve(_self:-ode,_self:-func); 
         o:=Object(solution_class,sol,false); 
         _self:-solutions_found:= [o]; 
         NULL; 
  end proc; 
end module: 
 
#------------------------- 
module first_order_ode_linear_class() 
  option object(ODE_base_class); 
  local f,g;  #ode of form  y'=f(x)*g(y) 
 
  #this method is not an object method. It is part of the module but does 
  #not have _self. It is called by the factory class to find if the ode 
  #is of this type first 
  export is_linear:=proc(ode::`=`,y::symbol,x::symbol)::truefalse; 
         RETURN(true); #for now 
  end proc; 
 
  export ModuleCopy::static:= proc(_self,proto::first_order_ode_linear_class,ode::`=`,y::symbol,x::symbol,$) 
      _self:-ode  := ode; 
      _self:-y  := y; 
      _self:-x  := x; 
      _self:-func  := _self:-y(_self:-x); 
      _self:-ode_order :=1; 
  end proc; 
 
  export dsolve::static:=proc(_self,$) 
         local sol,o; 
         sol:= :-dsolve(_self:-ode,_self:-func); 
         o:=Object(solution_class,sol,false); 
         _self:-solutions_found[1]:= [o]: 
  end proc: 
end module:
 

Example usage is

o:=ODE_factory_class:-make_ode(diff(y(x),x)=x,y(x)) 
o:-get_ode() 
                           d 
                          --- y(x) = x 
                           dx 
 
o:-dsolve(); 
o:-get_sol() 
                      [       1  2       ] 
                      [y(x) = - x  + c__1] 
                      [       2          ]
 

2 Differential equations

2.1 How to force dsolve to use specific method for solving?
2.2 How to find a particular solution to ODE?
2.3 How to find basis solutions for homogeneous ode?
2.4 How to solve a differential equation with initial conditions?
2.5 How to verify that the ODE solution given is correct?
2.6 How to know the type of ODE?
2.7 What packages to load for differential equations?
2.8 How to plot solution of differential equations?
2.9 On High precision. Using taylor to solve ODE
2.10 Obtain ODE in canonical coordinates (Lie symmetry)
2.11 How to use hint with symgen?
2.12 How to parse a single ode?
2.13 How to check if single ODE is valid?
2.14 How to check if an ode has \(y'\) in it?
2.15 How to check if an ode is linear ode?
2.16 How to find the order of an ode?
2.17 How to find the coefficients of a linear ode?
2.18 find order and degree of highest derivative
2.19 How to move all derivatives to one side in an equation?
2.20 How to obtains list of all derivatives in expression?
2.21 How to invert roles of dependent variable and independent variable in an ode?
2.22 How to find the indicial equation for an ODE?
2.23 How to write derivative
2.24 How to solve heat PDE in 1D in Maple 2017?
2.25 How to make Maple display diff(y(x),x) as y’(x) or as y’ ?
2.26 How to set boundary conditions for dsolve or pdsolve?
2.27 How force dsolve to use Lie?
2.28 How to do change of variables on the dependent variable for an ODE?
2.29 How to do change of variable on the independent variable for an ODE?
2.30 ODE change of variable on both dependent and independent variable?
2.31 How to make phase plot of first order ODE?
2.32 How to make phase plot of second order ODE?
2.33 How to move all terms with y to one side?
2.34 How to find all constants of integrations in an expression?
2.35 How to find if an ode is missing the dependent variable?

2.1 How to force dsolve to use specific method for solving?

To find what methods dsolve uses do

`dsolve/methods`[1] 
 
     [quadrature, linear, Bernoulli, separable, inverse_linear, 
      homogeneous, Chini, lin_sym, exact, Abel, pot_sym]
 

The above gives methods for first order ODEs. More are given by

`dsolve/methods`[1,'semiclass'] 
 
     [Riccati, inverse_Riccati, equivalent_to_Abel, 
      linearizable, linearizable_by_differentiation] 
 
`dsolve/methods`[1,'high_degree'] 
 
     [WeierstrassP, WeierstrassPPrime, JacobiSN, linearizable_by_differentiation, 
      missing, dAlembert, homogeneous_B, sym_implicit] 
 
`dsolve/methods`[1,"development"] 
 
     [linearizable_by_differentiation, linearizable, con_sym, 
      WeierstrassP, WeierstrassPPrime, equivalent_to_Abel, 
      Abel_AIR, special, Riccati_symmetries] 
 
`dsolve/methods`[1,extra] 
     [inverse_Riccati, Abel_AIL, `sym_pat/[F(x)*G(y),0]`, `sym_pat/[F(x),G(x)]`, 
      `sym_pat/[F(x),G(y)]`, `sym_pat/[F(x)+G(y),0]`, 
      `sym_pat/[F(x),G(x)*y+H(x)]`, sym_pat, exp_sym]
 

For example given a first order ode, we can ask dsolve to solve it using one of these methods as follows

restart; 
ode:=diff(y(x),x)=sqrt( sqrt( y(x) )* sin(x))/sqrt(y(x)) 
dsolve(ode,y(x),[`exact`]) 
 
     4/5*y(x)^(5/4) + Intat(-sqrt(sqrt(y(x))*sin(_a))/y(x)^(1/4), _a = x) + c__1 = 0
 

We can ask it to use symmetry with specific pattern as follows

restart; 
ode:=diff(y(x),x)=sqrt( sqrt( y(x) )* sin(x))/sqrt(y(x)) 
dsolve(ode,y(x),[`sym_pat/[F(x),G(x)*y+H(x)]`]) 
 
    c__1 + 4/5*y(x)^(3/2)*sqrt(sin(x))/sqrt(sqrt(y(x))*sin(x)) - Int(sqrt(sin(x)), x) = 0 
 
#or using short cut, same thing can be done as follows 
dsolve(ode,y(x),Lie) 
 
    c__1 + 4/5*y(x)^(3/2)*sqrt(sin(x))/sqrt(sqrt(y(x))*sin(x)) - Int(sqrt(sin(x)), x) = 0 
 
#another short cut for Lie is 
dsolve(ode,[`sym_pat`]) 
 
    c__1 + 4/5*y(x)^(3/2)*sqrt(sin(x))/sqrt(sqrt(y(x))*sin(x)) - Int(sqrt(sin(x)), x) = 0
 

To find all methods, and for higher order ode, we first use indices as follows

indices(`dsolve/methods`) 
 
    [high, linear_nonhomogeneous], 
    [1], 
    [1, high_degree], 
    [3, linear_homogeneous], 
    [2, "linear_homogeneous all methods"], 
    [2, "linear_homogeneous other"], 
    [2, linear_homogeneous], 
    [2, "special_functions"], 
    [3, linear_nonhomogeneous], 
    [2, "linear_homogeneous in Normal Form"], 
    [2, "development"], [2, "hypergeometric"], 
    [1, extra], 
    [high, linear_homogeneous], 
    [high, nonlinear], 
    [1, "special"], 
    [1, "development"], 
    [2, nonlinear], 
    [high, "development"], 
    [1, semiclass], 
    [3, nonlinear], 
    [2, linear_nonhomogeneous], 
    [2, "linear_homogeneous as given"], 
    [3, "development"]
 

Using the above, to find all methods for second order linear_homogeneous ode, the command is

`dsolve/methods`[2, "linear_homogeneous all methods"] 
 
    [quadrature, const_coeffs, Euler, linear_1, 
     `linear/missing_y`, Kovacic, special_functions, to_const_coeffs, 
      exact_linear, sym_1, Mathieu, MeijerG, Heun, HeunG, HeunC, HeunB, HeunD, 
      HeunT, mu_xy, equivalent_to_Bessel, to_Riccati, Bessel, elliptic, 
      Legendre, Whittaker, Kummer, cylindrical, hypergeometric, hypergeom1, 
      hypergeom2, Riemann, RNF, hypergeometricsols, rationalize_lode, with_periodic_functions] 
 
`dsolve/methods`[2, "linear_homogeneous other"] 
 
    [exact_linear, sym_1, to_const_coeffs, mu_xy, equivalent_to_Bessel, 
     to_Riccati, with_periodic_functions]
 

For example, given the ode \(y''+3 y'+ y=0\), we can now do

dsolve(ode,y(x)) 
     y(x) = c__1*exp(1/2*(sqrt(5) - 3)*x) + c__2*exp(-1/2*(3 + sqrt(5))*x) 
 
dsolve(ode,y(x),[`const_coeffs`]) 
    y(x) = c__1*exp(1/2*(sqrt(5) - 3)*x) + c__2*exp(-1/2*(3 + sqrt(5))*x) 
 
dsolve(ode,y(x),[`Kovacic`]) 
    y(x) = c__1*exp(1/2*(sqrt(5) - 3)*x) + c__2*exp(-1/2*(3 + sqrt(5))*x)
 

Not all methods ofcourse will work, as it depends on the ode type.

This function below lists all methods

ind:=indices(`dsolve/methods`); 
for item in ind do 
    cat("`dsolve/methods`",String(item)); 
    eval(parse(%)) 
od;
 

Which gives

"`dsolve/methods`[high, linear_nonhomogeneous]" 
    [quadrature, fully_exact_linear, linear_nonhomogeneous_[0,1], 
    exact_linear_nonhomogeneous, linear, exp_reduce] 
 
 
"`dsolve/methods`[1]" 
  [quadrature, linear, Bernoulli, separable, inverse_linear, 
    homogeneous, Chini, lin_sym, exact, Abel, pot_sym] 
 
"`dsolve/methods`[1, high_degree]" 
    [WeierstrassP, WeierstrassPPrime, JacobiSN, 
      linearizable_by_differentiation, missing, dAlembert, 
      homogeneous_B, sym_implicit] 
 
 
"`dsolve/methods`[3, linear_homogeneous]" 
     [quadrature, const_coeffs, Euler, fully_exact_linear, 
       to_const_coeffs, linear, exp_reduce, exact_linear, 
       with_periodic_functions] 
 
 
"`dsolve/methods`[2, "linear_homogeneous all methods"]" 
     [quadrature, const_coeffs, Euler, linear_1, linear/missing_y, 
      Kovacic, special_functions, to_const_coeffs, exact_linear, 
      sym_1, Mathieu, MeijerG, Heun, HeunG, HeunC, HeunB, HeunD, 
      HeunT, mu_xy, equivalent_to_Bessel, to_Riccati, Bessel, 
      elliptic, Legendre, Whittaker, Kummer, cylindrical, 
      hypergeometric, hypergeom1, hypergeom2, Riemann, RNF, 
      hypergeometricsols, rationalize_lode, with_periodic_functions] 
 
 
"`dsolve/methods`[2, "linear_homogeneous other"]" 
      [exact_linear, sym_1, to_const_coeffs, mu_xy, 
       equivalent_to_Bessel, to_Riccati, with_periodic_functions] 
 
"`dsolve/methods`[2, linear_homogeneous]" 
                      [linear_homogeneous] 
 
"`dsolve/methods`[2, "special_functions"]" 
    [Bessel, elliptic, Legendre, Kummer, Whittaker, hypergeometric, Mathieu] 
 
 
"`dsolve/methods`[3, linear_nonhomogeneous]" 
     [quadrature, fully_exact_linear, linear_nonhomogeneous_[0,1], 
      exact_linear_nonhomogeneous, linear, exp_reduce] 
 
 
"`dsolve/methods`[2, "linear_homogeneous in Normal Form"]" 
                           [linear_1] 
 
"`dsolve/methods`[2, "development"]" 
    [mu_xyp, mu_xyp2, mu_formal, mu_heuristic, exp_reduce, linear, 
     Bessel2, Whittaker_old, nonlinear_homogeneous, 
     exact_linear_nonhomogeneous, mu_y1, mu_x_y1, mu_y_y1, 
     mu_poly_yn, exp_sym, sym_pat, sym_8] 
 
 
"`dsolve/methods`[2, "hypergeometric"]" 
                      [hypergeom1, hyper3] 
 
"`dsolve/methods`[1, extra]" 
      [inverse_Riccati, Abel_AIL, sym_pat/[F(x)*G(y),0], 
       sym_pat/[F(x),G(x)], sym_pat/[F(x),G(y)], 
       sym_pat/[F(x)+G(y),0], sym_pat/[F(x),G(x)*y+H(x)], sym_pat, 
       exp_sym] 
 
 
"`dsolve/methods`[high, linear_homogeneous]" 
     [quadrature, const_coeffs, Euler, fully_exact_linear, 
       to_const_coeffs, linear, exp_reduce, exact_linear, 
       with_periodic_functions] 
 
"`dsolve/methods`[high, nonlinear]" 
     [linearizable_by_differentiation, linearizable, reducible, 
      exact_nonlinear, missing, mu_formal, lin_sym] 
 
"`dsolve/methods`[1, "special"]" 
                            [80, 81] 
 
"`dsolve/methods`[1, "development"]" 
     [linearizable_by_differentiation, linearizable, con_sym, 
      WeierstrassP, WeierstrassPPrime, equivalent_to_Abel, Abel_AIR, 
      special, Riccati_symmetries] 
 
 
"`dsolve/methods`[2, nonlinear]" 
    [Liouville, WeierstrassP, JacobiSN, linearizable, 
    linearizable_by_differentiation, mu_xy_2, missing, 
    mu_xyp2_dynamical_symmetries_fully_reducible, 
    mu_xyp_singularcases, sym_1, exact_nonlinear, reducible, 
    lin_sym, S-function, mu_xyp_generalcase, 
    mu_xyp2_dynamical_symmetries_not_fully_reducible] 
 
 
"`dsolve/methods`[high, "development"]" 
    [k25, RNF, mu_heuristic, MeijerG, nonlinear_homogeneous, 
     mu_poly_yn, exp_sym] 
 
"`dsolve/methods`[1, semiclass]" 
    [Riccati, inverse_Riccati, equivalent_to_Abel, linearizable, 
     linearizable_by_differentiation] 
 
 
"`dsolve/methods`[3, nonlinear]" 
   [linearizable_by_differentiation, linearizable, missing, 
     exact_nonlinear, reducible, mu_formal, lin_sym] 
 
 
"`dsolve/methods`[2, linear_nonhomogeneous]" 
     [quadrature, fully_exact_linear, linear_nonhomogeneous_[0,1], 
      linear_nonhomogeneous_[0,F(x)], linear_nonhomogeneous] 
 
"`dsolve/methods`[2, "linear_homogeneous as given"]" 
     [quadrature, const_coeffs, Euler, linear_1, linear/missing_y, 
      Kovacic, RNF, special_functions, MeijerG, Heun, 
      hypergeometricsols, rationalize_lode] 
 
"`dsolve/methods`[3, "development"]" 
     [k25, RNF, mu_heuristic, linear_patterns, MeijerG, 
      nonlinear_homogeneous, mu_y2, mu_poly_yn, exp_sym, pFq, 3F2, 
      2F2, 1F2, 0F2]
 

2.2 How to find a particular solution to ODE?

restart; 
ode:=diff(y(x),x)+y(x)^2*sin(x)-2*sin(x)/cos(x)^2 = 0; 
yp:=DETools:-particularsol(ode);
 

To step into the code, do

restart; 
ode:=diff(y(x),x)+y(x)^2*sin(x)-2*sin(x)/cos(x)^2 = 0; 
stopat(`DEtools/particularsol`); 
DETools:-particularsol(ode);
 

To print it do

print(`DEtools/particularsol`);
 

2.3 How to find basis solutions for homogeneous ode?

Use the output=basis option

ode:=diff(y(x),x$2)-x*diff(y(x),x)-x*y(x)=0; 
dsolve(ode,output=basis);
 

2.4 How to solve a differential equation with initial conditions?

To solve \[ y''-3y'+2y=10 e^{5 x} \] with \(y(0)=1,y'(0)=5\) do

eq1:= diff(y(x),x$2)-3*diff(y(x),x)+2*y(x)=10*exp(5*x); 
dsolve({eq1,y(0)=1,D(y)(0)=5},y(x)); 
 
Methods for second order ODEs: 
Trying to isolate the derivative d^2y/dx^2... 
Successful isolation of d^2y/dx^2 
--- Trying classification methods --- 
trying a quadrature 
trying high order exact linear fully integrable 
trying differential order: 2; linear nonhomogeneous with symmetry [0,1] 
trying a double symmetry of the form [xi=0, eta=F(x)] 
<- double symmetry of the form [xi=0, eta=F(x)] successful 
....
 

The above can also be written using D@@ notation, like this

eq:= (D@@2)(y)(x) - 3*D(y)(x) +2*y(x) = 10*exp(5*x); 
IC := y(0)=1,D(y)(0)=5; 
dsolve({eq,IC},y(x));
 

2.5 How to verify that the ODE solution given is correct?

use odetest and check if it gives zero.

eq1:= diff(diff(y(x),x),x)-3*diff(y(x),x)+2*y(x)=10*exp(5*x); 
ans:=dsolve({eq1,IC},y(x)); 
odetest(ans,eq1); 
 
                 0
 

2.6 How to know the type of ODE?

Maple can classify the ODE.

 
eq1:= diff(y(x),x$2)-3*diff(y(x),x)+2*y(x)=10*exp(5*x); 
R0 := DEtools['odeadvisor'](eq1,y(x)); 
 
          R0 := [[_2nd_order, _with_linear_symmetries]]
 

To get help on this type of ODE, do

DEtools['odeadvisor'](eq1,'help');
 

2.7 What packages to load for differential equations?

Use  with(DEtools);

2.8 How to plot solution of differential equations?

restart; 
eq1:= diff(y(x),x$2)-3*diff(y(x),x)+2*y(x)=10*exp(5*x); 
DEtools[DEplot](eq1,y(x),x=-2..5, [ [y(0)=0, D(y)(0)=0]], y=-3..3,linecolor=red);
 

To get a better plot, change the stepsize and independent variable range

restart; 
eq1:= diff(y(x),x$2)-3*diff(y(x),x)+2*y(x)=10*exp(5*x); 
DEtools[DEplot](eq1,y(x),x=-1..1,[[y(0)=0,D(y)(0)=0]],y=-3..3,stepsize=0.001,linecolor=red);
 

2.9 On High precision. Using taylor to solve ODE

From: Robert Israel (israel@math.ubc.ca) 
Subject: Re: given precision in Maple 
Newsgroups: comp.soft-sys.math.maple 
Date: 2003-07-16 20:19:06 PST 
 
Set Digits:= n and all calculations from this point will be done with n 
digits.  Mathematical functions will be correct to n digits as well (to 
the extent this is practical). 
 
If you want high-accuracy numerical ODE solutions, on the other hand, 
it's not so simple.  I think the best way is using the taylorseries 
method.  For example, consider the problem y' = y^2, y(1) = 1, where 
the exact solution y = 1/(2-x) has y(1.9) = 10. 
 
> Digits:= 30: 
  sol:= dsolve({D(y)(x)=y(x)^2, y(1) = 1}, y(x), numeric, 
            method=taylorseries, abserr=1e-25): 
  sol(1.9); 
 
          [x = 1.9, y(x) = 9.99999999999999999999999797691] 
 
> 10 - eval(y(x),%); 
 
                                       -23 
                            0.202309 10 
 
The other methods (in particular the default rkf45) do not give results 
anywhere near this good.
 

2.10 Obtain ODE in canonical coordinates (Lie symmetry)

How to obtain the ODE in canonical coordinates \(S(R)\) for an ode using Lie symmetry?

Here is a function I wrote with the help of Maple docs, that does that. For now, I am using first order ode. It takes the ode and the variables (coordinates) to use (\(R,S\)) is what normally used, and returns back the ODE in canonical coordinates. The ode should always be a quadrature for first order.

get_ODE_in_canonical:=proc(ode::`=`,y::symbol,x::symbol,S::symbol,R::symbol) 
  local infinitesimals,tr,itr,ODE; 
  infinitesimals := [DEtools:-symgen(ode)]; 
  if nops(infinitesimals)=0 then 
     RETURN(FAIL); 
  fi; 
  infinitesimals:=infinitesimals[1]; 
  tr  := DEtools:-canoni(infinitesimals,y(x),S(R)); 
  itr := op(1,[solve(tr,{x,y(x)})]); 
  ODE:= PDEtools:-dchange(itr,ode,[R,S(R)],simplify); 
  ODE:= op(solve(ODE,{diff(S(R),R)})); 
  RETURN(ODE); 
end proc:
 

Example

ODE := diff(y(x),x) = 1/x^2+y(x); 
get_ODE_in_canonical(ODE,y,x,S,R) 
 
    #diff(S(R), R) = 1/(exp(R)*R^2) 
 
ODE := diff(y(x),x) = x^2-2*x*y(x)+y(x)^2; 
get_ODE_in_canonical(ODE,y,x,S,R) 
   #diff(S(R), R) = 1
 

The following function is small change of the above. It accepts your choice of \(\xi ,\eta \) to use instead of the first one obtained by Maple. This makes it possible to experiment with different infinitesimals. Note that different infinitesimals will/can result in different canonical ODE, but the final solution for \(y(x)\) of course will always be the same after applying the transformation back to \(x,y\) coordinates.

restart; 
get_ODE_in_canonical_v2:=proc(ode::`=`,y::symbol,x::symbol,S::symbol,R::symbol,xi,eta) 
  local infinitesimals:=[xi,eta],tr,itr,ODE; 
  tr  := DEtools:-canoni(infinitesimals,y(x),S(R)); 
  itr := op(1,[solve(tr,{x,y(x)})]); 
  ODE:= PDEtools:-dchange(itr,ode,[R,S(R)],simplify); 
  ODE:= op(solve(ODE,{diff(S(R),R)})); 
  RETURN(simplify(ODE,symbolic)); 
end proc:
 

And now

ODE:=diff(y(x),x) = x^2-2*x*y(x)+y(x)^2; 
get_ODE_in_canonical_v2(ODE,y,x,S,R,1,1) 
 
    #diff(S(R), R) = 1/(R^2 - 1)
 

2.11 How to use hint with symgen?

Here is an example.

ode:=diff(y(x),x)=-y(x)^2/(exp(x)-y(x)) 
DEtools:-symgen(ode,y(x),HINT=[c1*x+c2*y+c3,c4*x+c5*y+c6]); 
 
      [_xi = 1, _eta = y]
 

Or

ode:=diff(y(x),x)=-y(x)^2/(exp(x)-y(x)) 
DEtools:-symgen(ode,y(x),HINT=[g(x),f(x)*y]); 
 
      [_xi = 1, _eta = y]
 

So not use y(x) but only y in the hint.

2.12 How to parse a single ode?

The following function takes in a single ode and parses it to verify it is valid. It returns back 3 values. The dependent variable, the indepenent variable and the ode order.

#added 12/8/2022. 
interface(warnlevel=4); 
kernelopts('assertlevel'=2): 
 
parse_single_ode:=proc(ode::`=`)::symbol,symbol,integer; 
#parses single ode. returns back  dep_var,indep_var,ode_order 
#throws exception when it detects parsing errors 
 
local func; 
local y,x; 
local dep_variables_found::list,item; 
local the_order; 
 
func:=PDEtools:-Library:-GetDepVars("not given",ode,onlydifferentiatedfunctions=true); 
if nops(func)=0 then 
   error ("not differential equation ",ode); 
fi; 
 
func := func[1]; 
 
if nops(func)<>1 then 
   error("Parsing error, dependent variable must contain one argument, found ", func); 
fi; 
 
y:=op(0,func); 
x:=op(1,func); 
 
#basic verification 
if not has(ode,y) then 
   error ("Supplied ode ",ode," has no ",y); 
fi; 
 
if not has(ode,x) then 
   error ("Supplied ode ",ode," has no ",x); 
fi; 
 
if not has(ode,func) then 
   error ("Supplied ode ",ode," has no ",func); 
fi; 
 
the_order := PDEtools:-difforder(ode,x); 
#if the_order=0 then 
#   error ("No derivative found in ",ode,". Input is not differential equation"); 
#fi; 
 
#note that the following call will also return y(x) if the input is not an ode 
#this will check that the dependent variable will show with SAME argument in the ode 
#i.e. if y(x) and y(t) show up in same ode, it will throw exception, which is what 
#we want. 
try 
   dep_variables_found := PDEtools:-Library:-GetDepVars([y],ode); 
catch: 
   error lastexception; 
end try; 
 
#now go over dep_variables_found and check the independent variable is same as x 
#i.e. ode can be y'(z)+y(z)=0 but function is y(x). 
for item in dep_variables_found do 
    if not type(item,function) then 
       error("Parsing error. Expected ",func," found ",item," in ode"); 
    else 
       if op(1,item) <> x then 
          error("Parsing error. Expected ",func," found ",item," in ode"); 
       fi; 
    fi; 
od; 
 
#now go over all indents in ode and check that y only shows as y(x) and not as just y 
#as the PDEtools:-Library:-GetDepVars([_self:-y],ode) code above does not detect this. 
#i.e. it does not check  y'(x)+y=0 
 
if numelems(indets(ode,identical(y))) > 0 then 
   error("Parsing error, Can not have ",y," with no argument inside ",ode); 
fi; 
 
return y,x,the_order; 
end proc:
 

To use do

y,x,the_order := parse_single_ode(diff(y(x),x)+y(x)=0);
 

An alternative to the above is to pass the dependent function itself as well as the ode. This is what I do myself in my ode solver. Like this

parse_single_ode:=proc(ode::`=`,func::function(name))::symbol,symbol,integer; 
#parses single ode. returns back  dep_var,indep_var,ode_order 
#throws exception when it detects parsing errors 
 
local y,x; 
local dep_variables_found::list,item; 
local the_order; 
 
    if nops(func)<>1 then 
       error("Parsing error, dependent variable must contain one argument, found ", func); 
    fi; 
 
    y:=op(0,func); 
    x:=op(1,func); 
 
    #basic verification 
    if not has(ode,y) then 
       error ("Supplied ode ",ode," has no ",y); 
    fi; 
 
    if not has(ode,x) then 
       error ("Supplied ode ",ode," has no ",x); 
    fi; 
 
    if not has(ode,func) then 
       error ("Supplied ode ",ode," has no ",func); 
    fi; 
 
    the_order := PDEtools:-difforder(ode,x); 
 
    #note that the following call will also return y(x) if the input is not an ode 
    #this will check that the dependent variable will show with SAME argument in the ode 
    #i.e. if y(x) and y(t) show up in same ode, it will throw exception, which is what 
    #we want. 
    try 
       dep_variables_found := PDEtools:-Library:-GetDepVars([y],ode); 
       #print("dep_variables_found=",dep_variables_found); 
    catch: 
       error lastexception; 
    end try; 
 
    #now go over dep_variables_found and check the independent variable is same as x 
    #i.e. ode can be y'(z)+y(z)=0 but function is y(x). 
    for item in dep_variables_found do 
        if not type(item,function) then 
           error("Parsing error. Expected ",func," found ",item," in ode"); 
        else 
           if op(1,item) <> x then 
              error("Parsing error. Expected ",func," found ",item," in ode"); 
           fi; 
        fi; 
    od; 
 
    #now go over all indents in ode and check that y only shows as y(x) and not as just y 
    #as the PDEtools:-Library:-GetDepVars([_self:-y],ode) code above does not detect this. 
    #i.e. it does not check  y'(x)+y=0 
 
    if numelems(indets(ode,identical(y))) > 0 then 
       error("Parsing error, Can not have ",y," with no argument inside ",ode); 
    fi; 
 
    return y,x,the_order; 
end proc:
 

To use do the same as before, but need to add \(y(x)\) as second argument. Like this

y,x,the_order := parse_single_ode(diff(y(x),x)+y(x)=0, y(x));
 

2.13 How to check if single ODE is valid?

This is very much the same as above, but this function returns True if the ode is syntactly valid, else false. Can be used just to check if the ode is valid before using it.

It takes in the ode and the function \(y(x)\) and returns either true or false.

#March 15, 2024 
 
export is_valid_single_ode :=proc(ode_in::`=`,func::function(name))::truefalse; 
    local x:=op(1,func),y:=op(0,func); 
    local ode:=ode_in; 
    local dep_variables_found::list; 
    local item; 
 
    if nops(func)<>1 then RETURN(false); fi; 
 
    if not has(ode,diff) then ode:=convert(ode,diff); fi; 
 
    if `or`(not has(ode,diff), not has(ode,x), not has(ode,func))  then 
       RETURN(false); 
    fi; 
 
    try 
        dep_variables_found := PDEtools:-Library:-GetDepVars([y],ode); 
    catch: 
       RETURN(false); 
    end try; 
 
    map( X-> `if`( `or`(not type(X,function), op(1,X) <> x) ,RETURN(false),NULL), 
       dep_variables_found 
    ); 
 
    #check there is no y on its own. Should always be y(x) 
    if nops(indets(ode,identical(y))) <> 0 then 
       RETURN(false); 
    fi; 
 
    RETURN(true); 
end proc:
 

Example usages

 
is_valid_single_ode(diff(y(x),x)+sin(x)=0, y(x) ) 
 
       true
 

To test, do

 
L:=[[3*(D@@2)(y)(x)+diff(y(x),x)+1=sin(x),y(x),true], 
                 [diff(y(x),x)^2=z,y(x),true], 
                 [diff(y(x),x)^2=z,y(x),true], 
                 [diff(y(x),x)^2=y(x)^2,y(x),true], 
                 [diff(y(x),x)=y(z)^2,y(x),false], 
                 [1/diff(y(x),x)=y(x)^2,y(x),true], 
                 [y(x)=1,y(x),false], 
                 [diff(y(x),z)=1,y(x),false], 
                 [D(y)(x)=1,y(x),true], 
                 [(D@@2)(y)(x)+diff(y(x),x)=sin(x),y(x),true], 
                 [diff(y(x),x)+diff(y(z),z)=1,y(x),false], 
                 [diff(y(x),x)+diff(g(z),z)=1,y(x),true] 
                 ]: 
 
map(X->evalb(is_valid_single_ode(X[1],X[2])=X[3]),L); 
if andmap(x->evalb(x=true),%) then 
   print("all tests passed"); 
else 
   print("WARNING, not all tests passed"); 
fi; 
 
#gives 
 
 [true, true, true, true, true, true, true, true, true, true, true, true] 
                       "all tests passed"
 

2.14 How to check if an ode has \(y'\) in it?

Suppose to want to check if an ode has \(y'(x)\) in it. We can not write has(ode, diff(y(x),x)) and see if this gives true or not, because this will also match \(y''\) and \(y'''\) and any higher order.

One way is to first convert the ode to D form and then use has on D(y)(x). This will not match \(y''\) anymore which is what we wanted, because \(y''\) becomes (D@@2)(y)(x) and so the check for first order diff will work as expected.

ode:=diff(y(x),x$2)+3*y(x)=0; 
ode:=convert(ode,D); 
has(ode,(D)(y)(x)) 
 
            #false
 

Another example

ode:=diff(y(x),x$2)+1/diff(y(x),x)=0; 
ode:=convert(ode,D); 
has(ode,(D)(y)(x)) 
 
            #true
 

The same thing if we wanted to check if the ode has \(y''\) or not.

ode:=diff(y(x),x$3)+diff(y(x),x)=0; 
ode:=convert(ode,D); 
has(ode,(D@@2)(y)(x)) 
 
       #false
 

2.15 How to check if an ode is linear ode?

ode:=diff(y(x),x$3)+ x*diff(y(x),x)+sin(x)=0; 
if has(DEtools:-odeadvisor(ode,y(x),['linear']),_linear) then 
   print("linear"); 
else 
   print("not linear ode"); 
fi
 

The above prints "linear"

ode:=y(x)*diff(y(x),x$3)+ x*diff(y(x),x)+sin(x)=0; 
if has(DEtools:-odeadvisor(ode,y(x),['linear']),_linear) then 
   print("linear"); 
else 
   print("not linear ode"); 
fi
 

The above prints "not linear ode". This works for any ode order.

2.16 How to find the order of an ode?

ode:=diff(y(x),x$3)+ x*diff(y(x),x)+sin(x)=0; 
PDEtools:-difforder(ode)
 

Gives \(3\).

2.17 How to find the coefficients of a linear ode?

Given a linear ode of the form \(A y''+B y' + C y= f(x)\) how to find \(A,B,C,f(x)\) ?

ode:=diff(y(x),x$3)+ x*diff(y(x),x)+99*y(x)=sin(x); 
L:=DEtools:-convertAlg(ode,y(x)); #this only works on linear ode's
 

Gives L := [[99, x, 0, 1], sin(x)].

\(L\) is a list. The second entry is \(f(x)\) and the first entry of \(L\) is a list which gives the coefficents of the ode. Notice they are ordered from lowest order to highest order of the ode. Since this is third order ode, then there are \(4\) entries. The first is the coefficient of \(y(x)\), the second is the coefficient of \(y'\) and the third is the coefficient of \(y''\) and the fourth entry is the coefficient of \(y'''\). Notice that coefficient of \(y''\) is zero since \(y''\) is missing from the ode.

This function DEtools:-convertAlg only works on linear ode’s. Therefore we need to check if the ode is linear first before using. How to check for linear ode is given above.

2.18 find order and degree of highest derivative

I need to find the degree of the highest derivative in an ode.

For example, if the input is \[ x \left (\frac {d^{2}}{d x^{2}}y \left (x \right )\right )^{5}+x \left (\frac {d^{2}}{d x^{2}}y \left (x \right )\right )^{2}+\left (\frac {d}{d x}y \left (x \right )\right ) y \left (x \right )+\left (\frac {d^{3}}{d x^{3}}y \left (x \right )\right )^{4} \sin \left (x \right )+5 y \left (x \right )+\sin \left (x \right )+\frac {d^{6}}{d x^{6}}r \left (x \right )+\left (\frac {d^{3}}{d x^{3}}y \left (x \right )\right )^{4} \cos \left (x \right ) = 0 \]

Then the degree for highest derivative of \(y\) w.r.t. \(x\) is 4. For \[ y''(x)^2 + y'(x)+y(x)=0 \] It will be 2.

This function returns the order and degree of such term.

 
find_all_derivatives_of_specific_order:=proc(expr,y::symbol,x::symbol,N::posint)::set; 
local t1,t2; 
 
if not has(expr,diff(y(x),x$N)) then 
   return {}; 
fi; 
 
t1  := identical(diff(y(x),x$N))^anything; 
t2  := identical(diff(y(x),x$N)); 
return indets[flat](expr,{t1,t2});  #MUST use flat 
 
end proc: 
 
get_order_and_degree_of_largest_derivative:=proc(expr,y::symbol,x::symbol)::integer,anything; 
local the_order,the_degree; 
local cand::set; 
local the_exponent; 
local item; 
 
    if not has(expr,diff(y(x),x)) then return 0,0; fi; 
    the_order := PDEtools:-difforder(expr,x); 
    cand       := find_all_derivatives_of_specific_order(expr,y,x,the_order); 
 
    if nops(cand)=0 then 
        the_degree := 0; 
    else 
        the_degree:=1; 
        for item in cand do 
            if type(item,`^`) then 
                the_exponent := op(2,item); 
                if type( the_exponent,symbol) or not type( the_exponent,numeric) then #assume larger 
                    the_degree :=the_exponent; 
                else 
                    if type(the_degree,symbol) or not  type( the_exponent,numeric) then 
                        next; 
                    else 
                        if op(2,item)>the_degree then 
                            the_degree := op(2,item); 
                        fi; 
                    fi; 
                fi; 
            else 
                if type(the_degree,symbol) then 
                    next; 
                else 
                    if the_degree=0 then 
                        the_degree := 1; 
                    fi; 
                fi; 
            fi; 
        od; 
    fi; 
 
    return the_order,the_degree; 
end proc:
 

And now it can be called as follows

ode:=diff(y(x),x$2)+diff(y(x),x)+y(x)=0; 
get_order_and_degree_of_largest_derivative(lhs(ode),y,x) 
 
    #2,1 
 
ode:=diff(y(x),x$2)*diff(y(x),x$2)^4+diff(y(x),x)+y(x)=0; 
get_order_and_degree_of_largest_derivative(lhs(ode),y,x) 
 
   #2,5 
 
ode:=y(x)=0; 
get_order_and_degree_of_largest_derivative(lhs(ode),y,x) 
 
   #0,0 
 
ode:=diff(y(x),x)^n=0; 
get_order_and_degree_of_largest_derivative(lhs(ode),y,x) 
 
   #1,n
 

2.19 How to move all derivatives to one side in an equation?

Given \[ a \left (\frac {d}{d x}y \left (x \right )\right )+b \left (\frac {d^{2}}{d x^{2}}y \left (x \right )\right )+x +\cos \left (x \right )+y \left (x \right )+c \left (\frac {d}{d x}y \left (x \right )\right )^{2} = \sin \left (x \right ) \] How to move all terms with derivative to LHS side and everything else to RHS?

ode:= a*diff(y(x),x)+b*diff(y(x),x$2)+x+cos(x)+y(x)+c*diff(y(x),x)^2=sin(x); 
ode:=lhs(ode)-rhs(ode); 
LHS,RHS:=selectremove(has,ode,'diff'); 
new_ode:=LHS=-RHS;
 

\[ a \left (\frac {d}{d x}y \left (x \right )\right )+b \left (\frac {d^{2}}{d x^{2}}y \left (x \right )\right )+c \left (\frac {d}{d x}y \left (x \right )\right )^{2} = -x -\cos \left (x \right )-y \left (x \right )+\sin \left (x \right ) \]

2.20 How to obtains list of all derivatives in expression?

I had a need to find all derivatives of form  diff(y( anything), anything) in an ode so to check that \(y\) argument is not different among them.

For an example, given \[ a \left (\frac {d^{2}}{d x^{2}}y \left (x \right )\right ) \left (\frac {d^{3}}{d x^{3}}y \left (x \right )\right )-\sqrt {1+\left (\frac {d^{2}}{d x^{2}}y \left (x \right )\right )^{2}}+\frac {1}{\sin \left (\frac {d^{5}}{d x^{5}}y \left (x \right )\right )} = \frac {d}{d z}y \left (z \right )+{\mathrm e}^{y \left (x \right )+\frac {d}{d x}y \left (x \right )}+\frac {d}{d x}r \left (x \right ) \]

the result should be \[ \left [\begin {array}{c} \frac {d^{5}}{d x^{5}}y \left (x \right ) \\ \frac {d^{4}}{d x^{4}}y \left (x \right ) \\ \frac {d^{3}}{d x^{3}}y \left (x \right ) \\ \frac {d^{2}}{d x^{2}}y \left (x \right ) \\ \frac {d}{d x}y \left (x \right ) \\ \frac {d}{d z}y \left (z \right ) \end {array}\right ] \]

One issuse is how to check for diff and also check that the dependent variable is \(y\) so as not to pick other dependent variables such as \(z\) in this example. This was done by converting diff to D otherwise it will not work.

restart; 
expr:=a*diff(y(x),x$2)*diff(y(x),x$3)-sqrt(1+ diff(y(x),x$2)^2)+1/sin(diff(y(x),x$5))=diff(y(z),z)+exp(y(x)+diff(y(x),x))+diff(r(x),x); 
 
#this finds all derivatives 
list_of_diffs:=indets(expr,'satisfies'(s->op(0,s)='diff' and op([0,1],convert(s,D))=y)); 
 
#This finds all dependent variables 
list_of_diffs:=convert(list_of_diffs,list); 
list_of_indep_variables   := map(x->PDEtools:-Library:-GetIndepVars(x)[-1],list_of_diffs); 
 
#this converts it to set. If the ODE is valid, then the list_of_indep_variables should 
#have one entry $x$ in it and nothing else. 
list_of_indep_variables   := convert(ListTools:-Flatten(list_of_indep_variables),set); 
 
if nops(list_of_indep_variables)>1 then 
  error( cat("Only one independent variable expected in differential form, found ", 
                   convert(list_of_indep_variables,string)) ); 
fi; 
 
if list_of_indep_variables[1]<>x then 
   error( cat("Independent variable expected in differential form not same as independent variable in function given ", 
      convert(list_of_indep_variables,string)) ); 
fi;
 

Another option instead of doing all the above is to do this

expr:=a*diff(y(x),x$2)*diff(y(x),x$3)-sqrt(1+ diff(y(x),x$2)^2)+1/sin(diff(y(x),x$5))=diff(y(z),z)+exp(y(x)+diff(y(x),x))+diff(r(x),x); 
try 
   PDEtools:-Library:-GetDepVars([y(x)],expr); 
catch 
  error "functions with name [y] having different dependency: [[y(x), y(z)]]" 
end try;
 

The function  PDEtools:-Library:-GetDepVars([y(x)],expr) checks that only \(y(x)\) dependency shows up. It throws an error otherwise. So if an error is thrown, then this means \(y\) shows up with different independent variables.

2.21 How to invert roles of dependent variable and independent variable in an ode?

Sometimes it is useful to invert an ode. i.e. make the independent variable the dependent variable, and the dependent variable the independent. For example, given \[ 1+\left (\frac {x}{y \left (x \right )}-\sin \left (y \left (x \right )\right )\right ) \left (\frac {d}{d x}y \left (x \right )\right ) = 0 \] We want the ode to become \[ -\sin \left (y \right ) y +y \left (\frac {d}{d y}x \left (y \right )\right )+x \left (y \right ) = 0 \]

This can be done as follows

restart; 
 
ode:=1+ (x/y(x)-sin(y(x) ))*diff(y(x),x)=0; 
tr:={x=u(t),y(x)=t}; 
ode:=PDEtools:-dchange(tr,ode); 
ode:=eval(ode,[t=y,u=x]); 
ode:=simplify(ode);
 

\[ \frac {-\sin \left (y \right ) y +y \left (\frac {d}{d y}x \left (y \right )\right )+x \left (y \right )}{y \left (\frac {d}{d y}x \left (y \right )\right )} = 0 \]

In this case, we can get rid of the denomator, but this is a manual step for now.

ode:=numer(lhs(ode))=0;
 

\[ -\sin \left (y \right ) y +y \left (\frac {d}{d y}x \left (y \right )\right )+x \left (y \right ) = 0 \]

The above can now be solved more easily for \(x(y)\) than solving the orignal ode for \(y(x)\).

2.22 How to find the indicial equation for an ODE?

For say Bessel ode of order zero:

eq:= x^2*diff(y(x),x$2)+x*diff(y(x),x)+x^2*y(x)=0; 
DEtools[indicialeq](eq,x,0,y(x)); 
          #x^2 = 0
 

The third argument above is the singularity point of interest. So we have two roots, both zero. These are now used for finding the power series solution \(y(x)\) if needed.

Another example, is Bessel of order 1

eq:= x^2*diff(y(x),x$2)+x*diff(y(x),x)+(x^2-1)*y(x)=0; 
DEtools[indicialeq](eq,x,0,y(x)); 
          #x^2-1 = 0
 

2.23 How to write derivative

To write \(y'(x)=x\), one way is diff(y(x),x)=x and another is D(y)(x)=x. To write \(y''(x)=x\), one way is diff(y(x),x$2)=x and another is (D@@2)(y)(x)=x.

To convert from one form to another use convert(eq,diff) or convert(eq,D)

2.24 How to solve heat PDE in 1D in Maple 2017?

to solve \(\frac {\partial u(x,t)}{\partial t}=k \frac {\partial ^2 u(x,t)}{\partial x^2}\) with homogeneous dirichlet boundary conditions \(u(0,t)=0,u(L,t)=0\) the commands are

restart; 
pde:=diff(u(x,t),t)=k*diff(u(x,t),x$2); 
bc:=u(0,t)=0,u(L,t)=0; 
sol:=pdsolve([pde,bc]) assuming 0<L:
 

Which gives

\[ u \left ( x,t \right ) =\sum _{{\it \_Z1}=1}^{\infty }{\it \_C1} \left ( {\it \_Z1} \right ) \sin \left ( {\frac {\pi \,{\it \_Z1}\,x}{L}} \right ) {{\rm e}^{-{\frac {k{\pi }^{2}{{\it \_Z1}}^{2}t}{{L}^{2}}}}} \]

Which can be made more readable as follows

sol:=algsubs(_Z1=n,sol): 
sol:=algsubs(Pi*n/L=lambda(n),sol);
 

\[ u \left ( x,t \right ) =\sum _{n=1}^{\infty }{\it \_C1} \left ( n \right ) \sin \left ( x\lambda \left ( n \right ) \right ) {{\rm e}^{-kt \left ( \lambda \left ( n \right ) \right ) ^{2}}} \]

For homogeneous Neumann B.C., at \(x=0\), let \(\frac {\partial u}{\partial x}=0\) and at \(x=L\) let \(u(L,t)=0\), the solution it gives looks different than my hand solution

restart; 
pde:=diff(u(x,t),t)=k*diff(u(x,t),x$2); 
bc:=D[1](u)(0,t)=0,u(L,t)=0; 
pdsolve([pde,bc]) assuming 0<L;
 

It gives

\[ u \left ( x,t \right ) ={\it \_C3}\,{\it \_C2}\, \left ( {{\rm e}^{1/4\,{\frac {2\,i\pi \,xL-k{\pi }^{2}t}{{L}^{2}}}}}+{{\rm e}^{-1/4\,{\frac {\pi \, \left ( 2\,ixL+k\pi \,t \right ) }{{L}^{2}}}}} \right ) \]

I need to look more into the above and see if this comes out to be the same as my hand solution.

Another example, with initial conditions now given

restart; 
pde:=diff(u(x,t),t)=k*diff(u(x,t),x$2); 
bc:=D[1](u)(0,t)=0,u(L,t)=0; 
ic:=u(x,0)=f(x); 
sol:=pdsolve([pde,bc,ic],u(x,t)) assuming 0<L; 
sol1:=algsubs(_Z2=n,sol);
 

The result is

\[ u \left ( x,t \right ) =\sum _{n=1}^{\infty } \left ( 2\,{\frac {1}{L}{{\rm e}^{-1/4\,{\frac {k{\pi }^{2}t \left ( 1+2\,n \right ) ^{2}}{{L}^{2}}}}}\cos \left ( 1/2\,{\frac {\pi \,x \left ( 1+2\,n \right ) }{L}} \right ) \int _{0}^{L}f \left ( x \right ) \cos \left ( 1/2\,{\frac {\pi \,x \left ( 1+2\,n \right ) }{L}} \right ) \,{\rm d}x} \right ) \]

Another example

restart; 
pde:=diff(u(x,t),t)=k*diff(u(x,t),x$2); 
bc:=D[1](u)(0,t)=0,u(L,t)=0; 
ic:=u(x,0)=3*sin(Pi*x/L)-sin(3*Pi*x/L); 
sol:=pdsolve([pde,bc,ic],u(x,t)) assuming 0<L; 
sol1:=algsubs(_Z2=n,sol);
 

\[ u \left ( x,t \right ) =\sum _{n=1}^{\infty }768\,{\frac {1}{\pi \, \left ( 16\,{n}^{4}+32\,{n}^{3}-136\,{n}^{2}-152\,n+105 \right ) }{{\rm e}^{-1/4\,{\frac {k{\pi }^{2}t \left ( 1+2\,n \right ) ^{2}}{{L}^{2}}}}}\cos \left ( 1/2\,{\frac {\pi \,x \left ( 1+2\,n \right ) }{L}} \right ) } \]

Another example

restart; 
pde:=diff(u(x,t),t)=k*diff(u(x,t),x$2); 
bc:=u(0,t)=0,u(L,t)=0; 
ic:=u(x,0)=3*sin(Pi*x/L)-sin(3*Pi*x/L); 
sol:=pdsolve([pde,bc,ic],u(x,t)) assuming 0<L;
 

\[ u \left ( x,t \right ) =\sin \left ( {\frac {\pi \,x}{L}} \right ) {{\rm e}^{-9\,{\frac {{\pi }^{2}kt}{{L}^{2}}}}} \left ( -2\,\cos \left ( 2\,{\frac {\pi \,x}{L}} \right ) +3\,{{\rm e}^{8\,{\frac {{\pi }^{2}kt}{{L}^{2}}}}}-1 \right ) \]

The above answer seems wrong. There is not even a summation in it. It is different from my hand solution. Look more into it.

2.25 How to make Maple display diff(y(x),x) as y’(x) or as y’ ?

Add this

expr:=diff(y(x),x); 
Typesetting:-Settings(typesetprime=true, prime=x):
 

The above will display the expression as \(y'(x)\). To make it now show the \(x\) do

expr:=diff(y(x),x); 
Typesetting:-Settings(typesetprime=true, prime=x): 
Typesetting:-Suppress(y(x));
 

Now it will show the expression as just \(y'\). For all the above to work, make sure you have Typesetting level set to Extended in the GUI.

This is done inside Tools->Options->Display menu.

To clear all the above Typesetting, do restart or do Typesetting:-Unsuppress(y(x))

2.26 How to set boundary conditions for dsolve or pdsolve?

The Maple syntax for seeting initial and boundary conditions is very confusing, as compared to Mathematica, which seems to me to be simpler. So I wrote this to remind me of the syntax each time.

For PDE, assuming dependent variable is \(u(x,t)\) then

Conditions Maple code
\(u(0,t)=0\)  u(0,t)=0
\(\frac {\partial u}{\partial x}=0\) at \(x=0\) D[1](u)(0,t)=0
\(\frac {\partial ^2 u}{\partial x^2}=0\) at \(x=0\) D[1,1](u)(0,t)=0
\(\frac {\partial ^3 u}{\partial x^3}=0\) at \(x=0\) D[1,1,1](u)(0,t)=0
\(\frac {\partial u}{\partial t}=0\) at \(t=0\) D[2](u)(x,0)=0
\(\frac {\partial ^2 u}{\partial t^2}=0\) at \(t=0\) D[2,2](u)(x,0)=0
\(\frac {\partial ^3 u}{\partial t^3}=0\) at \(t=0\) D[2,2,2](u)(x,0)=0

Notice the syntax for the last one above. It is (D[1]@@2)(u)(0,t)=0 and not (D@@2)[1](u)(0,t)=0

For an ODE, assuming dependent variable is \(y(x)\) then the syntax is

Conditions Maple code
\(y(0)=0\)  y(0)=0
\(\frac {dy}{dx}=0\) at \(x=0\) D(y)(0)=0
\(\frac {d^2 y}{d x^2}=0\) at \(x=0\) (D@@2)(y)(0)=0

2.27 How force dsolve to use Lie?

Use dsolve(ode,Lie)

To find symmetries, do

DEtools:-symgen(ode,y(x),HINT=[c__1+c__2*x+c__3*y,c__4+c__5*x+c__6*y])

or just

DEtools:-symgen(ode,y(x))

To debug it do

stopat(`ODEtools/symgen`); before calling dsolve or DEtools:-symgen

2.28 How to do change of variables on the dependent variable for an ODE?

given an ode

\[ y \left (x \right ) = \left (\frac {d}{d x}y \left (x \right )\right )^{3} y \left (x \right )^{2}+2 x \left (\frac {d}{d x}y \left (x \right )\right ) \]

do change of variable \(u(x)=y(x)^2\)

restart; 
ode:=y(x)=diff(y(x),x)^3*y(x)^2+2*x*diff(y(x),x); 
new_ode:=PDEtools:-dchange({y(x)=sqrt(u(x))},ode,{u});
 

\[ \sqrt {u \left (x \right )} = \frac {\left (\frac {d}{d x}u \left (x \right )\right )^{3}}{8 \sqrt {u \left (x \right )}}+\frac {x \left (\frac {d}{d x}u \left (x \right )\right )}{\sqrt {u \left (x \right )}} \]

2.29 How to do change of variable on the independent variable for an ODE?

given an ode

\[ \frac {d^{2}}{d t^{2}}y \left (t \right )+y \left (t \right ) = 2 t \]

do change of variable \(t=\tau + \pi \)

restart; 
ode:=diff(y(t),t$2)+y(t)=2*t; 
PDEtools:-dchange({t=tau+Pi},ode,known={t},unknown={tau},params=Pi)
 

\[ \frac {d^{2}}{d \tau ^{2}}y \left (\tau \right )+y \left (\tau \right ) = 2 \tau +2 \pi \]

it is important to use  params=Pi. Watch what happens if we do not do that

restart; 
ode:=diff(y(t),t$2)+y(t)=2*t; 
PDEtools:-dchange({t=tau+Pi},ode,known={t},unknown={tau});
 

\[ \frac {d^{2}}{d \tau ^{2}}y \left (\tau , \pi \right )+y \left (\tau , \pi \right ) = 2 \tau +2 \pi \] Which is not what we want.

2.30 ODE change of variable on both dependent and independent variable?

This verifies solution given in https://math.stackexchange.com/questions/3477732/can-t-see-that-an-ode-is-equivalent-to-a-bessel-equation Where a change of variables on \[ \xi ^2 \frac {d^2\eta }{d\xi ^2} + \xi \frac {d\eta }{d\xi } - (\xi ^2+n^2)\eta =0 \] Was made using \[ \eta =\frac {y}{x^\alpha }, \quad \xi =\beta x^\gamma , \] To produce the ode \[ \frac {d^2y}{dx^2} - \frac {(2\alpha -1)}{x}\frac {dy}{dx} - (\beta ^2 \gamma ^2 x^{2\gamma -2}+\frac {n^2\gamma ^2-\alpha ^2}{x^2})y=0. \] In Maple the above is done using

restart; 
ode := zeta^2*diff(eta(zeta),zeta$2) + zeta*diff(eta(zeta),zeta) - (zeta^2 + n^2)*eta(zeta) = 0; 
the_tr:={zeta=beta*x^gamma,eta(zeta)=y(x)/x^alpha}; 
PDEtools:-dchange(the_tr,ode,{y(x),x},'known'={eta(zeta)},'uknown'={y(x)},'params'={alpha,beta,gamma}); 
simplify(%); 
numer(lhs(%))=0; 
simplify(numer(lhs(%))/(x^(1-alpha)))=0; 
numer(lhs(%))=0; 
collect(%,[y(x),diff(y(x),x),diff(y(x),x$2)]);
 

Which gives \[ \left (-\gamma ^{2} x^{-1+2 \gamma } \beta ^{2} x -\gamma ^{2} n^{2}+\alpha ^{2}\right ) y \left (x \right )+\left (-2 \alpha x +x \right ) \left (\frac {d}{d x}y \left (x \right )\right )+x^{2} \left (\frac {d^{2}}{d x^{2}}y \left (x \right )\right ) = 0 \]

Here is another example. Here to make change of variables to polar coordinates by making \(x=r \cos \theta \) and \(y=r \sin \theta \) The ode is \[ \frac {y-x y'}{\sqrt {1+(y')^2}}=x^2+y^2 \]

In Maple

restart; 
ode := (y(x)-x*diff(y(x),x))/sqrt(1+ diff(y(x),x)^2) = x^2+y(x)^2; 
the_tr:={x=r(t)*cos(t),y(x)=r(t)*sin(t)}; 
PDEtools:-dchange(the_tr,ode,{r(t),t},'known'={y(x)},'uknown'={r(t)});
 

Which gives

\[ \frac {r \left (t \right ) \sin \left (t \right )-\frac {r \left (t \right ) \cos \left (t \right ) \left (\left (\frac {d}{d t}r \left (t \right )\right ) \sin \left (t \right )+r \left (t \right ) \cos \left (t \right )\right )}{\left (\frac {d}{d t}r \left (t \right )\right ) \cos \left (t \right )-r \left (t \right ) \sin \left (t \right )}}{\sqrt {1+\frac {\left (\left (\frac {d}{d t}r \left (t \right )\right ) \sin \left (t \right )+r \left (t \right ) \cos \left (t \right )\right )^{2}}{\left (\left (\frac {d}{d t}r \left (t \right )\right ) \cos \left (t \right )-r \left (t \right ) \sin \left (t \right )\right )^{2}}}} = r \left (t \right )^{2} \left (\cos ^{2}\left (t \right )\right )+r \left (t \right )^{2} \left (\sin ^{2}\left (t \right )\right ) \]

Here is another example. Where we want to change \(R(r)\) to \(y(x)\) everywhere

\[ \frac {d^{2}}{d r^{2}}R \left (r \right )+\frac {d}{d r}R \left (r \right )+R \left (r \right ) = 0 \]

restart; 
ode:=diff(R(r),r$2)+diff(R(r),r)+R(r)=0; 
the_tr:={r=x,R(r)=y(x)}; 
PDEtools:-dchange(the_tr,ode,{y(x),x},'known'={R(r),r},'uknown'={y(x),x});
 

\[ \frac {d^{2}}{d x^{2}}y \left (x \right )+\frac {d}{d x}y \left (x \right )+y \! \left (x \right ) = 0 \]

The format of the transformation is old_independent_variable=new_independent_variable and old_dependent_variable=new_dependent_variable

2.31 How to make phase plot of first order ODE?

Make a phase plot of \[ y'(x) = \sqrt {y(x)^{2}-1} \]

The phase plot has \(x\) on the x axis and has \(y\) on the y axis. It shows the family of solutions for different initial conditions.

restart; 
ode:=diff(y(x), x) = sqrt(y(x)^2 - 1); 
DEtools:-DEplot(ode,y(x),x=-2..2,y=1..3)
 

To show specific solution curve that passes via some initial conditions such as \(y(0)=2\) then do

restart; 
ode:=diff(y(x), x) = sqrt(y(x)^2 - 1); 
DEtools:-DEplot(ode,y(x),x=-2..2,y=1..3,[y(0)=2])
 

2.32 How to make phase plot of second order ODE?

Make a phase plot of \[ \frac {d^{2}}{d t^{2}}x \left (t \right )+\frac {\frac {d}{d t}x \left (t \right )}{2}+x \left (t \right ) = u \left (t \right ) \]

By plotting \(x(t)\) vs \(x'(t)\) without solving the ODE.

restart; 
alias(DS=DynamicSystems): 
ode := diff(x(t),t$2) +1/2*diff(x(t),t)+ x(t) = u(t); 
sys:=DS:-DiffEquation(ode,'outputvariable'=[x(t)],'inputvariable'=[u(t)]); 
sys0:=DS:-StateSpace(sys); 
eq1:=diff(x1(t),t)=sys0:-a[1,..].Vector([x1(t),x2(t)]); 
eq2:=diff(x2(t),t)=sys0:-a[2,..].Vector([x1(t),x2(t)]); 
DEtools:-DEplot([eq1,eq2],[x1(t),x2(t)],t=0..35,[[x1(0)=1,x2(0)=1]],x1=-2..2,x2=-2..2, 
        numpoints=200, linecolor=black, axes=boxed);
 

2.33 How to move all terms with y to one side?

Given ode where \(y\) is dependent variable and \(x\) is independent variable, how to normalize the ode so that all terms with \(y(x)\) are on left side and everything is on the RHS? THis makes it easier to see what the forcing function is. i.e. we want to write the ode as

\[ a y''+b y'+c y= f(x) \]

Here is function to do this. This works for any ode or any expression.

export move_y_x_to_each_side:=proc(expr_in::`=`,y::symbol,x::symbol)::`=`; 
    local expr:=expr_in; 
    local stuff_with_y,stuff_not_with_y; 
    expr  := lhs(expr)-rhs(expr); 
    if not expr::`+` then 
      RETURN(expr_in); 
    fi; 
    expr  := numer(normal(expr)); 
    stuff_with_y,stuff_not_with_y := selectremove(has,expr,y); 
    stuff_not_with_y := convert(stuff_not_with_y,diff); 
 
    RETURN(stuff_with_y = -stuff_not_with_y); 
end proc:
 

Examples

ode:=diff(y(x),x$2)+x*y(x)+sin(x)-3=tan(x); 
expr:=move_y_x_to_each_side(ode,y,x); 
 
        # diff(y(x), x, x) + x*y(x) = 3 - sin(x) + tan(x)
 

And

ode:=diff(y(x),x$2)+x*y(x)+sin(x)-3=tan(x)+y(x)+Pi; 
expr:=move_y_x_to_each_side(ode,y,x); 
 
        # diff(y(x), x, x) + x*y(x) - y(x) = 3 - sin(x) + tan(x) + Pi
 

And

ode:=move_y_x_to_each_side(diff(x(t),t$2)/x(t) + diff(x(t),t)/x(t) - F(t)/x(t) =1,x,t) 
expr:=move_y_x_to_each_side(ode,x,t); 
 
        # diff(x(t), t, t) + diff(x(t), t) - x(t) = F(t)
 

2.34 How to find all constants of integrations in an expression?

Sometimes I make solution for an ode with constants of integrations. These must be of form _Cn where \(n\) is positive integer, such as _C1 or _C5. Now to integrate this solution once more, need to come up with new constant that does not already show in the solution.

To find current constants in an expression use this

sol:=x-_C1*x+_C3*x^2; 
indets(sol,And(symbol, suffixed(_C, nonnegint))); 
 
         {_C1, _C3}
 

Therefore, one way to make a new constant of integration, is to find the largest numbered one and increase by one. Like this

restart; 
sol:=x-_C1*x+_C3*x^2; 
myconstants:=indets(sol,And(symbol, suffixed(_C, nonnegint))); 
map(X->String(X),myconstants); 
map(X->X[3..],%); 
map(X->:-parse(X),%); 
n:=max(%); 
new_constant:=_C||(n+1);
 

2.35 How to find if an ode is missing the dependent variable?

Sometimes it is usefull to know if ode is missing \(y(x)\) as this allows one to do the substitution \(y'=u\) and reduce the order of the ode.

This is a function which takes in an ode and returns true if it is missing \(y(x)\) or false otherwise.

#added 2/18/2024. checks if an ode is missing y 
export is_ode_missing_y:=proc(ode_in::`=`,y::symbol,x::symbol)::truefalse; 
local ode:=ode_in; 
local ode_order::posint := PDEtools:-difforder(ode_in); 
local N::posint; 
 
    ode:=lhs(ode)-rhs(ode); 
    ode:=convert(ode,D); 
    for N from 1 to ode_order do 
        ode:=eval(ode,[(D@@N)(y)(x)=Y||N]); 
    od; 
 
    RETURN(not has(ode,y(x))); 
 
end proc;
 

To use do

restart; 
ode:=diff(y(x),x$2)+diff(y(x),x$4)+x=0; 
is_ode_missing_y(ode,y,x); 
 
       #true 
 
ode:=diff(y(x),x$2)+diff(y(x),x$4)+x=sin(x)+y(x); 
is_ode_missing_y(ode,y,x); 
 
      #false
 

3 Symbolic manipulation, expression filtering, structured types

3.1 on maple data types
3.2 how to extract elements from a list based on some selection?
3.3 how to test if all elements of a matrix are integers?
3.4 How to select terms with sqrt or radical
3.5 How to find all csgn() and replace them by 1
3.6 How to find symbols inside csgn() in an expression?
3.7 How to replace all abs(expr) by expr
3.8 How to obtain list of all occurances of some function in an expression?
3.9 How to replace \(\ln (|x|)\) with \(\ln (x)\) in an expression?
3.10 How to find all signum functions in expression and simplify it?
3.11 How to replace all signum functions in expression by 1?
3.12 How to find if some function is present in an expression
3.13 How to find all functions in an expression?
3.14 find all functions except builtin math functions
3.15 How to obtain a list of all arguments of function?
3.16 Find functions whose first argument is \(z\)
3.17 Find functions whose second argument is \(t\)?
3.18 How to use select with own type to find subexpressions?
3.19 How to write structured types to match some expressions?
3.20 select only indexed variables from an expression
3.21 Given an expression, how to find all variables and functions in it?
3.22 How to check if an expression is integer, when it has symbols in it?
3.23 How to find parameters such as \(\pi \) in an expression?
3.24 How to find all derivatives \(y'(x)\) in an expression?
3.25 How combine log terms?
3.26 Find position in a list of items that are not numeric
3.27 How to change \(\arctan (y,x)\) to \(\arctan \left (\genfrac {}{}{}{}{y}{x}\right )\) in an expression
3.28 How to find all symbols that represent variables in an expression?
3.29 How to change first argument of function?
3.30 How to change last argument of function?
3.31 How to remove last argument of function?
3.32 How to find a pattern inside an expression
3.33 How to find parts of a Sum?
3.34 find if sequence or list is inside another and the indecies
3.35 find if some type inside some expression and its location
3.36 How to replace generic function inside derivative?
3.37 Examples how to match types
3.38 On the order of terms when using indents

3.1 on maple data types

See http://www.maplesoft.com/applications/view.aspx?SID=1533&view=html&L=G

3.2 how to extract elements from a list based on some selection?

use select. For example

>restart; 
>my_list:=[1,3.4,3+I,5]; 
>select(x->evalb(Im(x)=0),my_list); 
        [1, 3.4, 5]
 

3.3 how to test if all elements of a matrix are integers?

restart; 
m:=Matrix( [[1.3,2,3],[3,4,4] ]); 
matrixTestQ := proc(m::Matrix) 
 
    local r,c,i,j; 
 
    (r,c):=LinearAlgebra[Dimensions](m); 
    for i from 1 to r do 
       for j from 1 to c do 
           if( not evalb( whattype(m[i,j]) = integer) )  then 
               return(false); 
            end if; 
        end do; 
    end do; 
 
    return true; 
end proc; 
 
>matrixTestQ(m); 
 
               false
 

I am sure there is a better way than the above. Need to find out.

3.4 How to select terms with sqrt or radical

Given

\[ 3+x +\sqrt {-4 a c +b^{2}}+\sin \left (y \right )+x^{3} \sqrt {39}+\sqrt {\cos }x \] Find terms that are sqrt. Use indets

restart; 
expr_with_radical:= 3+x+sqrt(b^2-4*a*c)+sin(y)+x^3*sqrt(39)+sqrt(cos(x));| 
indets(expr_with_radical, algebraic^fraction)
 

\( \{ \sqrt {39}, \sqrt {-4 a c +b^{2}}, \sqrt {\cos }x \} \)

Alternative is to use type radical

restart; 
expr_with_radical:= 3+x+sqrt(b^2-4*a*c)+sin(y)+x^3*sqrt(39)+sqrt(cos(x));| 
indets(expr_with_radical, radical)
 

\( \{ \sqrt {39}, \sqrt {-4 a c +b^{2}}, \sqrt {\cos }x \} \)

3.5 How to find all csgn() and replace them by 1

I wanted to simplify an expression which could have csgn() in it, and find all the arguments.

\[ \frac {1+\mathrm {csgn} \left (a \right ) a}{3 \mathrm {csgn} \left (b \right ) b} \] One way is

restart; 
expr:=(1+csgn(a)*a)/(3*csgn(b)*b): 
expr:=subsindets(expr,'specfunc( anything, csgn )',f->1);
 

\[ \frac {1+a}{3 b} \]

3.6 How to find symbols inside csgn() in an expression?

Given  sol:=1/2*2^(1/2)*csgn(x)*x*csgn(y);  how to find all symbols inside csgn which will be \(x,y\) in this case?

restart; 
sol:=1/2*2^(1/2)*csgn(x)*x*csgn(y); 
indets(sol,'specfunc( anything, csgn )'); 
vars:=subsindets(%,'specfunc( anything, csgn )',f->op(f))
 

Gives {x, y}

Now if we want to simplify the above solution by assuming that all variables inside vars are positive, how to do that?

restart; 
sol:=1/2*2^(1/2)*csgn(x)*x*csgn(y); 
indets(sol,'specfunc( anything, csgn )'); 
vars:=subsindets(%,'specfunc( anything, csgn )',f->op(f)); 
simplify(sol) assuming op(map2(`<`,0,vars))
 

Gives \(\frac {\sqrt {2}\, x}{2}\). Notice in the above the use of op(map2(`<`,0,vars)), this will generate the sequence 0 < x, 0 < y automatically. op is needed otherwise the result will be {0 < x, 0 < y} which will give syntax error when passed to assuming

Ofcourse, it would have been also possible to just write

simplify(sol) assuming positive;
 

And get the same result. But sometimes we might want to specify which variables are to be assumed positive and not all of them at once in the expression.

3.7 How to replace all abs(expr) by expr

I wanted to replace  |expr| by (expr)

One way is

restart; 
expr:=u(x) = _C1*exp(-3*x^(1/3)*sqrt(c))*(3*x^(1/3)*sqrt(c) + 1) + _C2*exp(3*x^(1/3)*sqrt(c))*abs(-1 + 3*x^(1/3)*sqrt(c));
 

\[ u \left (x \right ) = \textit {\_C1} \,{\mathrm e}^{-3 x^{\frac {1}{3}} \sqrt {c}} \left (3 x^{\frac {1}{3}} \sqrt {c}+1\right )+\textit {\_C2} \,{\mathrm e}^{3 x^{\frac {1}{3}} \sqrt {c}} {| -1+3 x^{\frac {1}{3}} \sqrt {c}|} \]

expr:=subsindets(expr,'specfunc( anything, abs )',f->op(f));
 

\[ u \left (x \right ) = \textit {\_C1} \,{\mathrm e}^{-3 x^{\frac {1}{3}} \sqrt {c}} \left (3 x^{\frac {1}{3}} \sqrt {c}+1\right )+\textit {\_C2} \,{\mathrm e}^{3 x^{\frac {1}{3}} \sqrt {c}} \left (-1+3 x^{\frac {1}{3}} \sqrt {c}\right ) \]

3.8 How to obtain list of all occurances of some function in an expression?

For an example, How to find list of all \(\ln \) functions in this expression?

\[ \ln \left ({| x +1|}\right )+2 x \ln \left (x \right )+\sin \left (x \right ) \]

restart; 
expr:=ln(abs(x+1))+2*x*ln(x)+sin(x); 
tmp := indets(expr,'specfunc(anything,ln)'); 
 
          #   tmp := {ln(x), ln(abs(x + 1))}
 

To pick only \(\ln \) functions which has \(abs\) inside them anywhere, replace the above with

restart; 
expr:=ln(abs(x+1))+2*x*ln(x)+sin(x); 
lis:=indets(expr,'specfunc(anything,ln)'); 
select(Z->has(Z,abs),lis) 
 
          #   tmp := {ln(abs(x + 1))}
 

Or, better alternative to the above is

restart; 
expr:=ln(abs(x+1))+2*x*ln(x)+sin(x); 
indets(expr,'specfunc( satisfies(u->has(u,abs)) ,ln  )'); 
 
          #   tmp := {ln(abs(x + 1))}
 

3.9 How to replace \(\ln (|x|)\) with \(\ln (x)\) in an expression?

Given

\[ \sin \left (x \right )+\ln \left ({| x |}\right )+\ln \left (x +\frac {{| y |}}{\sqrt {{| x +3|}}}\right )+\ln \left (x^{3}\right )+\cos \left ({| x |}\right ) \]

How to remove the absolute, the ones only inside each \(\ln \) in the above expression?

restart; 
expr:=sin(x)+ln(abs(x))+ln(x+abs(y)/sqrt(abs(x+3)))+ln(x^3)+cos(abs(x)); 
expr:=evalindets(expr,'specfunc(ln)',f->evalindets(f,'specfunc(abs)',f->op(1,f))) 
 
#   sin(x) + ln(x) + ln(x + y/sqrt(x + 3)) + ln(x^3) + cos(abs(x))
 

\[ \sin \left (x \right )+\ln \left (x \right )+\ln \left (x +\frac {y}{\sqrt {x +3}}\right )+\ln \left (x^{3}\right )+\cos \left ({| x |}\right ) \]

3.10 How to find all signum functions in expression and simplify it?

Given

\[ -\frac {\left (\ln \left (\frac {\left (b +\sqrt {b^{2}+y \left (x \right )^{2}}\, \mathrm {signum}\left (b \right )\right ) b}{y \left (x \right )}\right )+\ln \left (2\right )\right ) \mathrm {signum}\left (b \right )}{b} = \mathit {\_C1} +\frac {-\ln \left (a \right )+\ln \left (x \right )-\ln \left (a +\sqrt {a^{2}+x^{2}}\, \mathrm {signum}\left (a \right )\right )-\ln \left (2\right )}{{| a |}} \]

How to find all arguments of signum and simplify the above by assuming they are all positive?

restart; 
expr:=-(ln((b + sqrt(b^2 + y(x)^2)*signum(b))*b/y(x)) + ln(2))*signum(b)/b = _C1 + (-ln(a) + ln(x) - ln(a + sqrt(a^2 + x^2)*signum(a)) - ln(2))/abs(a); 
lis:=indets(expr,'specfunc(anything,signum)'); 
assum:=convert(map(x->op(1,x)>0,lis),list); 
simplify(expr,assume=assum);
 

\[ \frac {-\ln \left (b \right )-\ln \left (\frac {b +\sqrt {b^{2}+y \left (x \right )^{2}}}{y \left (x \right )}\right )-\ln \left (2\right )}{b} = \frac {\mathit {\_C1} a -\ln \left (a \right )-\ln \left (a +\sqrt {a^{2}+x^{2}}\right )+\ln \left (x \right )-\ln \left (2\right )}{a} \]

3.11 How to replace all signum functions in expression by 1?

Given

\[ -\frac {\left (\ln \left (\frac {\left (b +\sqrt {b^{2}+y \left (x \right )^{2}}\, \mathrm {signum}\left (b \right )\right ) b}{y \left (x \right )}\right )+\ln \left (2\right )\right ) \mathrm {signum}\left (b \right )}{b} = \mathit {\_C1} +\frac {-\ln \left (a \right )+\ln \left (x \right )-\ln \left (a +\sqrt {a^{2}+x^{2}}\, \mathrm {signum}\left (a \right )\right )-\ln \left (2\right )}{{| a |}} \]

How to replace all signum by 1?

restart; 
expr:=-(ln((b + sqrt(b^2 + y(x)^2)*signum(b))*b/y(x)) + ln(2))*signum(b)/b = _C1 + (-ln(a) + ln(x) - ln(a + sqrt(a^2 + x^2)*signum(a)) - ln(2))/abs(a); 
evalindets(expr, 'specfunc(anything,signum)', f -> 1);
 

\[ -\frac {\ln \left (\frac {\left (b +\sqrt {b^{2}+y \left (x \right )^{2}}\right ) b}{y \left (x \right )}\right )+\ln \left (2\right )}{b} = \textit {\_C1} +\frac {-\ln \left (a \right )+\ln \left (x \right )-\ln \left (a +\sqrt {a^{2}+x^{2}}\right )-\ln \left (2\right )}{{| a |}} \]

3.12 How to find if some function is present in an expression

Given expression \(3 \sin \left (x \right )+t +3 f \left (x , t\right ) t +g \left (x , t\right )\) find if it contains function \(f()\).

Use indets with specfunc(f)

restart; 
expr := 3*sin(x)+t+3*f(x,t)*t+g(x,t); 
res  := indets(expr, specfunc(f)); 
if numelems(res)<>0 then 
   print("Found f(x,t)"); 
else 
   print("could not find f(x,t)"); 
fi;
 

"Found f(x,t)"

3.13 How to find all functions in an expression?

Given expression \(3 \sin \left (x \right )+t +3 f \left (x , t\right ) t +g \left (x , t\right )\) find all functions, if any, in the expression.

Use indets with function

restart; 
expr := 3*sin(x)+t+3*f(x,t)*t+g(x,t); 
res  := indets(expr,function); 
if numelems(res)<>0 then 
   print("Found these functions",res); 
else 
   print("could not find any function)"); 
fi;
 

"Found these functions", f(x, t), g(x, t), sin(x)

3.14 find all functions except builtin math functions

Given expression \(3 \sin \left (x \right )+t +3 f \left (x , t\right ) t +g \left (x , t\right )\) find all functions, if any, in the expression but exclude the math functions such as \(\sin \) in the above.

restart; 
expr := 3*sin(x)+t+3*f(x,t)*t+g(x,t); 
res  := indets(expr, And( function, Not(typefunc(mathfunc)))); 
if numelems(res)<>0 then 
   print("Found these functions",res); 
else 
   print("could not find any function)"); 
fi;
 

"Found these functions", f(x, t), g(x, t)

3.15 How to obtain a list of all arguments of function?

use op

restart; 
op(1..,f(x,t)) 
 
    x, t
 

Note that op(0,f(x,t)) finds the function name.

3.16 Find functions whose first argument is \(z\)

restart; 
expr := 3*sin(z)+t+3*f(z,t,y)*t+g(x,t); 
res  := indets(expr, patfunc(identical(z), anything)); 
if numelems(res)<>0 then 
   print("Found these functions",res); 
else 
   print("could not find any function)"); 
fi;
 

gives

"Found these functions", f(z, t, y), sin(z)

3.17 Find functions whose second argument is \(t\)?

expr := 3*sin(z)+t+3*f(z,t,y)*t+g(x,t); 
res  := indets(expr, patfunc(anything, identical(t), anything)); 
if numelems(res)<>0 then 
   print("Found these functions",res); 
else 
   print("could not find any function)"); 
fi;
 

gives

"Found these functions", f(z, t, y), g(x, t)

3.18 How to use select with own type to find subexpressions?

Given expression such as \(3+(1+x)\sin x\) or \(3+(1+x)\sin ^2 x\) use select to find any  polynomial * sin^n  subexpressions.

restart; 
mytype_1 := ''`*`'( {polynom(And(algebraic, satisfies(u -> not has(u, I))),x), 
                     Or( 'specfunc(sin)'^integer, 'specfunc(sin)') 
                    } 
                  )'; 
select(type, 3+(1+x)*sin(x),mytype_1); 
select(type, 3+(1+x)*sin(x)^2,mytype_1);
 

Gives

                         (1 + x) sin(x) 
 
                                      2 
                        (1 + x) sin(x)
 

3.19 How to write structured types to match some expressions?

3.19.1 type for \(\sin ^m(x)\cos ^n(x)\)
restart; 
my_type:=''`*`'( { Or('specfunc'(sin),'specfunc'(sin)^Or(integer,rational)), 
                   Or('specfunc'(cos),'specfunc'(cos)^Or(integer,rational))})'; 
 
type(sin(x)^2*cos(x)^3,my_type); 
type(sin(x)^2*cos(x),my_type); 
type(sin(x)*cos(x),my_type); 
type(cos(x)*sin(x)^(1/2),my_type); 
 
 
           true 
           true 
           true 
           true
 

I could not find a way to avoid writing  Or('specfunc'(sin),'specfunc'(sin)^Or(integer,rational) in order to match both \(\sin x\) and \(\sin ^2 x\). For these things, I find Mathematica patterns more flexiable. The above can be done as follows in Mathematica

 
ClearAll[x,n,m,any] 
patt=any_.*Sin[_]^n_. * Cos[_]^m_. 
MatchQ[Sin[x]^2*Cos[2*x]^3,patt] 
MatchQ[Sin[x]^2*Cos[x],patt] 
MatchQ[Sin[x]*Cos[x],patt] 
MatchQ[Cos[x]*Sin[x],patt] 
 
      True 
      True 
      True 
      True
 

In Mathematica  n_.  says basically to match \(\sin x\) or \(\sin ^2 x\) since the dot says to match zero or more. So no need to duplicate things as I did above in Maple. There might be a way to do the same in Maple using structured type, but I could not find it. In General, I find patterns in Mathematica more flexible and easier to use for this sort of thing. Maple has patmatch command, but not as easy to use as Patterns in Mathematica.

3.20 select only indexed variables from an expression

use indets with type 'indexed'

expr:=16*a[3]+6*a[1]; 
terms:=indets(expr,'indexed'); 
 
         terms := {a[1], a[3]} 
 
#to find maximum index, then do 
 
map(x->op(x),terms) 
 
      {1, 3}
 

3.21 Given an expression, how to find all variables and functions in it?

Given say \(\frac {d^{2}}{d x^{2}}y \left (x \right )+n \left (\frac {d}{d x}y \left (x \right )\right )+3 = \sin \left (x \right )\) how to find all variables and functions in it, not including math functions such as \(\sin x\)?

So the result should be \(n,x,y(x)\).

ode:=diff(y(x),x$2)+n*diff(y(x),x)+3=sin(x); 
vars:=indets(ode, Or(  And(symbol,Not(constant)), And(function,Not(typefunc(mathfunc)) ) )) 
 
#gives 
#           vars := {n, x, diff(y(x), x), diff(y(x), x, x), y(x)}
 

I still need to work on excluding derivatives from the search.

3.22 How to check if an expression is integer, when it has symbols in it?

I had case where I needed to check if something is integer or not. The problem is that the result had a symbol \(n\) in it. I need a way to tell Maple that to check if the result can be an integer given that \(n\) is also an integer.

Using type does not work, since can’t use assumptions. One way is to use coulditbe as follows

restart; 
expr:=n-1+2*m; 
vars:=indets(expr,And(symbol,Not(constant))); 
coulditbe(expr,integer) assuming op(map(Z->Z::integer,vars)) 
 
#  true
 

In the above indets(expr,And(symbol,Not(constant))) picks all variables in the expression, and assuming op(map(Z->Z::integer,vars)) makes assumption that each is integer.

3.23 How to find parameters such as \(\pi \) in an expression?

Use

restart; 
expr:=4*Pi+sin(x); 
indets(expr,And(name,constant)) 
 
        {Pi}
 

3.24 How to find all derivatives \(y'(x)\) in an expression?

Given an expression such as

\[ \sin \left (x \right )+\left (\frac {d}{d x}y \left (x \right )\right )^{3}+\left (\frac {d}{d x}y \left (x \right )\right ) \ln \left (y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )^{2}\right )+\sqrt {\frac {d}{d x}y \left (x \right )}+\frac {x}{\left (\frac {d}{d x}y \left (x \right )\right )^{7}} \]

Find all \(y'(x)\) for any power that show up, so the result should be

\[ \left \{\frac {1}{\left (\frac {d}{d x}y \left (x \right )\right )^{7}}, \left (\frac {d}{d x}y \left (x \right )\right )^{2}, \left (\frac {d}{d x}y \left (x \right )\right )^{3}, \sqrt {\frac {d}{d x}y \left (x \right )}, \frac {d}{d x}y \left (x \right ) \right \} \]

Use indets with type identical(diff(y(x),x))^anything is used. But must use the flat option to work correctly.

restart; 
 
expr:= y(x)*diff(y(x),x)^(1/3)+sin(x)*diff(y(x),x)^3 + z*diff(y(x),x)*ln(y(x)*diff(y(x),x)^2) +diff(y(x),x)^(1/2) + x/diff(y(x),x)^7; 
t1:=identical(diff(y(x),x))^anything; 
t2:=identical(diff(y(x),x)); 
indets[flat](expr, 'Or'(  t1, t2 ));
 

gives

{diff(y(x), x)^(1/3), 1/diff(y(x), x)^7, diff(y(x), x)^2, diff(y(x), x)^3, diff(y(x), x), sqrt(diff(y(x), x))}
 

Without using flat it will given wrong result. For example

restart; 
 
expr:=diff(y(x),x)^2; 
t1:=identical(diff(y(x),x))^anything; 
t2:=identical(diff(y(x),x)); 
indets(expr, 'Or'(  t1, t2 ));
 

Gives

{diff(y(x), x)^2, diff(y(x), x)}
 

You see, it has extra diff(y(x), x) showing up. Adding flat it gives

restart; 
 
expr:=diff(y(x),x)^2; 
t1:=identical(diff(y(x),x))^anything; 
t2:=identical(diff(y(x),x)); 
indets[flat](expr, 'Or'(  t1, t2 ));
 

Now it gives

{diff(y(x), x)^2}
 

Which is the correct result.

3.25 How combine log terms?

To go from \(\ln (A B)\) to \(\ln A + \ln B\) need to use simplify with ln option but add assumptions that one of the terms is positive. Else it will not do it

restart; 
simplify(ln(A*B),ln); # no change 
simplify(ln(A*B),ln) assuming A>0;  # ln(A) + ln(B) 
simplify(ln(A*B),ln) assuming B>0;  # ln(A) + ln(B)
 

To go from \(\ln ( \frac {A}{B})\) to \(\ln A - \ln B\) need to use simplify with ln option but add assumptions that \(B>0\).

restart; 
simplify(ln(A/B),ln);  # no change 
simplify(ln(A/B),ln) assuming A>0;  # ln(A) + ln(1/B) 
simplify(ln(A/B),ln) assuming B>0;  # do this: -ln(B) + ln(A)
 

To go from \(\ln A + \ln B\) to \(\ln (A B)\) need to use combine with assumptions that either \(A\) or \(B\) is positive, else it will not do it.

restart; 
combine( ln(A) + ln(B),ln); # no change 
combine( ln(A) + ln(B),ln) assuming A>0; # ln(A*B) 
combine( ln(A) + ln(B),ln) assuming B>0  # ln(A*B)
 

To go from \(\ln A - \ln B\) to \(\ln \frac {A}{B} \) need to use combine with assumptions that either \(B\) is positive.

restart; 
combine( ln(A) - ln(B),ln);  #no change 
combine( ln(A) - ln(B),ln) assuming A>0;  # -ln(B/A) 
combine( ln(A) - ln(B),ln) assuming B>0   # use this  ln(A/B)
 

3.26 Find position in a list of items that are not numeric

Given list such as [1,2,3,4,5,x,y,8,9,Pi] find position of elements that are not numeric. In this case the answer should be [6,7,10]

restart; 
lis:=[1,2,3,4,5,x,y,8,9,Pi]; 
lis2:=select(x->not(type(x,numeric)),lis); 
map(x->ListTools:-Search(x,lis),lis2) 
 
    [6,7,10]
 

I could not find a way to do it using one command like with Mathematica. The first command above uses select to first find non numeric entries. The second command ListTools:-Search then find the index/position.

Maple’s ListTools:-Search should really have a version that allows one to select the element directly. Something like this

lis:=[1,2,3,4,5,x,y,8,9,Pi]; 
ListTools:-Search(x->not(type(x,'numeric')),lis,all)
 

3.27 How to change \(\arctan (y,x)\) to \(\arctan \left (\frac {y}{x}\right )\) in an expression

I needed to do this as I was translating Maple code to Sagemath. Where sagemath supports \(\arctan \) with only one argument.

Given an expression such as \[ x +\arctan \left (y,x \right )+\sin \left (x \right ) \] Convert it to \[ x +\arctan \left (\frac {y}{x}\right )+\sin \left (x \right ) \]

evalindets(expr,'specfunc(arctan)',f->`if`(nops(f)=2,arctan(op(1,f)/op(2,f)),f))
 

3.28 How to find all symbols that represent variables in an expression?

Given an expression such as \(a+\sin (x) + \pi + y+ f(r) \) how to find all symbols in it, which will be \(a,x,y,r\)?

One way

expr:=a+sin(x) + Pi + y+ f(r); 
indets(expr,And(symbol,Not(constant)));
 

Another is

expr:=a+sin(x) + Pi + y+ f(r); 
indets(expr,assignable(name));
 

Both give {a, r, x, y}

3.29 How to change first argument of function?

This question came up in https://mathematica.stackexchange.com/questions/274535/replacing-only-variables-in-specific-locations-with-replace-all where the user wanted to I would like to replace R0 with r, but only in the arguments of functions.

The input they had is line = R0*f[R0,x] + R0^2*42*D[g[R0,x],x]

In Maple, this can be done as follows, which I think is a easier thanks to Maple’s strong type system.

line := R0*f(R0,x) + R0^2*42*diff(g(R0,x),x); 
evalindets(line, 'patfunc(identical(R0),anything)', Z-> subsop(1 = r, Z ));
 

Which gives

\[ \mathit {R0} f \left (r , x\right )+42 \mathit {R0}^{2} \left (\frac {\partial }{\partial x}g \left (r , x\right )\right ) \]

3.30 How to change last argument of function?

Given an expression which contains some different functions each with different number of arguments. Suppose we want to change only the last argument of each function if the last argument is \(x\), and change it to say \(x^2\).

how to do that?

Hence given R0*f(R0,x)+42*R0^2*D[2](g)(R0,x)+h(x,y,z,r,x) we want to change it to R0*f(R0,x^2)+42*R0^2*D[2](g)(R0,x^2)+h(x,y,z,r,x^2).

expr:= R0*f(R0,x) + R0^2*42*diff(g(R0,x),x)+ h(x,y,z,r,x); 
evalindets(expr, 'patfunc[reverse](anything,identical(x))', Z-> subsop(-1 = x^2, Z ));
 

Note the use of  subsop(-1 = x^2, Z ) where \(-1\) means the last entry in the argument of the function. This is basically same as last example, but uses patfunc[reverse] instead of just patfunc

3.31 How to remove last argument of function?

This is the same example as the above, but now we want to remove the last argument instead of changing it.

Hence, given an expression which contains some different functions each with different number of arguments. Suppose we want to remove only the last argument of each function if the last argument is \(x\).

how to do that?

Hence given R0*f(R0,x)+h(x,y,z,r,x) we want to change it to R0*f(R0)+h(x,y,z,r).

expr:=R0*f(R0,x)+h(x,y,z,r,x) 
evalindets(expr, 'patfunc[reverse](anything,identical(x))', Z-> subsop(-1 = NULL, Z ));
 

Be careful using the above on expression that has diff(f(y,x),x) as this will give 0, because we basically removed the variable of the differentiation.

3.32 How to find a pattern inside an expression

Given \(3 \,{\mathrm e}^{x}+\sin \left (a \,{\mathrm e}^{x}\right ) f\left ({\mathrm e}^{5 x}\right )\) how to find all terms with pattern anything*exp ? If we do this

expr:=3*exp(x)+sin(a*exp(x))*f(exp(5*x)); 
indets(expr,`&*`(anything,'specfunc(exp)')); 
 
#        {a*exp(x), 3*exp(x)}
 

We see it did not find exp(5*x) this is because there is nothing multiplying the exp function. To find this we add Or like this to count for both cases

expr:=3*exp(x)+sin(a*exp(x))*f(exp(5*x)); 
indets[flat](expr,Or( `&*`(anything,'specfunc(exp)'), 'specfunc(exp)' )) 
 
 
#        {a*exp(x), 3*exp(x), exp(5*x)}
 

The flat option is needed, as without it this will be the result

expr:=3*exp(x)+sin(a*exp(x))*f(exp(5*x)); 
indets(expr,Or( `&*`(anything,'specfunc(exp)'), 'specfunc(exp)' )) 
 
#       {a*exp(x), 3*exp(x), exp(x), exp(5*x)}
 

3.33 How to find parts of a Sum?

Given an inert Sum such as \[ r = \sum _{n=2}^{\infty } a_{n -1} x^{n -1} \] How to obtain the body of the sum, the index variable, the lower starting value and the upper limit?

r:=Sum