Internal
problem
ID
[8388]
Book
:
Own
collection
of
miscellaneous
problems
Section
:
section
5.0
Problem
number
:
1
Date
solved
:
Sunday, November 10, 2024 at 03:41:30 AM
CAS
classification
:
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
Solve
Time used: 80.334 (sec)
This is missing independent variable second order ode. Solved by reduction of order by using substitution which makes the dependent variable \(y\) an independent variable. Using
Then
Hence the ode becomes
Which is now solved as first order ode for \(p(y)\).
The ode \(p^{\prime } = \frac {A \,y^{{2}/{3}}}{p}\) is separable as it can be written as
Where
Integrating gives
Solving for \(p\) gives
For solution (1) found earlier, since \(p=y^{\prime }\) then we now have a new first order ode to solve which is
Unable to integrate (or intergal too complicated), and since no initial conditions are given, then the result can be written as
Singular solutions are found by solving
for \(y\). This is because we had to divide by this in the above step. This gives the following singular solution(s), which also have to satisfy the given ODE.
For solution (2) found earlier, since \(p=y^{\prime }\) then we now have a new first order ode to solve which is
Unable to integrate (or intergal too complicated), and since no initial conditions are given, then the result can be written as
Singular solutions are found by solving
for \(y\). This is because we had to divide by this in the above step. This gives the following singular solution(s), which also have to satisfy the given ODE.
Will add steps showing solving for IC soon.
The solution
was found not to satisfy the ode or the IC. Hence it is removed.
Summary of solutions found
Time used: 83.138 (sec)
Multiplying the ode by \(y^{\prime }\) gives
Integrating the above w.r.t \(x\) gives
Which is now solved for \(y\). Solving for the derivative gives these ODE’s to solve
Now each of the above is solved separately.
Solving Eq. (1)
Unable to integrate (or intergal too complicated), and since no initial conditions are given, then the result can be written as
Singular solutions are found by solving
for \(y\). This is because we had to divide by this in the above step. This gives the following singular solution(s), which also have to satisfy the given ODE.
Solving Eq. (2)
Unable to integrate (or intergal too complicated), and since no initial conditions are given, then the result can be written as
Singular solutions are found by solving
for \(y\). This is because we had to divide by this in the above step. This gives the following singular solution(s), which also have to satisfy the given ODE.
Will add steps showing solving for IC soon.
The solution
was found not to satisfy the ode or the IC. Hence it is removed.
Summary of solutions found
`Methods for second order ODEs: --- Trying classification methods --- trying 2nd order Liouville trying 2nd order WeierstrassP trying 2nd order JacobiSN differential order: 2; trying a linearization to 3rd order trying 2nd order ODE linearizable_by_differentiation trying 2nd order, 2 integrating factors of the form mu(x,y) trying differential order: 2; missing variables `, `-> Computing symmetries using: way = 3 -> Calling odsolve with the ODE`, (diff(_b(_a), _a))*_b(_a)-A*_a^(2/3) = 0, _b(_a), HINT = [[_a, (5/6)*_b]]` *** Sublevel 2 *** symmetry methods on request `, `1st order, trying reduction of order with given symmetries:`[_a, 5/6*_b]
Solving time : 0.086
(sec)
Leaf size : 61
dsolve(diff(diff(y(x),x),x) = A*y(x)^(2/3), y(x),singsol=all)
Solving time : 0.117
(sec)
Leaf size : 75
DSolve[{D[y[x],{x,2}]==A*y[x]^(2/3),{}}, y[x],x,IncludeSingularSolutions->True]