Internal
problem
ID
[8877]
Book
:
Own
collection
of
miscellaneous
problems
Section
:
section
3.0
Problem
number
:
19
Date
solved
:
Sunday, March 30, 2025 at 01:45:25 PM
CAS
classification
:
[[_3rd_order, _with_linear_symmetries]]
Time used: 4.242 (sec)
The ode can be normalized and rewritten as Euler ode.
This is Euler ODE of higher order. Let
Substituting these back into
gives
Which simplifies to
And since
Simplifying gives the characteristic equation as
Solving the above gives the following roots
This table summarises the result
root | multiplicity | type of root |
| | complex conjugate root |
| | real root |
The solution is generated by going over the above table. For each real root
The fundamental set of solutions for the homogeneous solution are the following
This is higher order nonhomogeneous Euler type ODE. Let the solution be
Where
Now the particular solution to the given ODE is found
Let the particular solution be
Where
Where
Substituting the fundamental set of solutions
The determinant simplifies to
Now we determine
Now we are ready to evaluate each
Now that all the
Hence
Therefore the particular solution is
Therefore the general solution is
ode:=x^4*diff(diff(diff(y(x),x),x),x)+x^3*diff(diff(y(x),x),x)+x^2*diff(y(x),x)+x*y(x) = x; dsolve(ode,y(x), singsol=all);
Maple trace
Methods for third order ODEs: --- Trying classification methods --- trying a quadrature trying high order exact linear fully integrable trying differential order: 3; linear nonhomogeneous with symmetry [0,1] trying high order linear exact nonhomogeneous trying differential order: 3; missing the dependent variable checking if the LODE is of Euler type <- LODE of Euler type successful Euler equation successful
Maple step by step
ode=x^4*D[y[x],{x,3}]+x^3*D[y[x],{x,2}]+x^2*D[y[x],x]+x*y[x]== x; ic={}; DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
from sympy import * x = symbols("x") y = Function("y") ode = Eq(x**4*Derivative(y(x), (x, 3)) + x**3*Derivative(y(x), (x, 2)) + x**2*Derivative(y(x), x) + x*y(x) - x,0) ics = {} dsolve(ode,func=y(x),ics=ics)