Internal
problem
ID
[9123]
Book
:
Second
order
enumerated
odes
Section
:
section
1
Problem
number
:
52
Date
solved
:
Wednesday, March 05, 2025 at 07:26:52 AM
CAS
classification
:
[[_2nd_order, _missing_x]]
Solve
Factoring the ode gives these factors
Now each of the above equations is solved in turn.
Solving equation (1)
Solving for
Solving gives
Solving equation (2)
Time used: 226.095 (sec)
This is missing independent variable second order ode. Solved by reduction of order by using substitution which makes the dependent variable
Then
Hence the ode becomes
Which is now solved as first order ode for
Factoring the ode gives these factors
Now each of the above equations is solved in turn.
Solving equation (1)
Solving for
Solving gives
Solving equation (2)
The ode has the form
Where
Solving for
To be able to solve as separable ode, we have to now assume that
Under the above assumption the differential equations become separable and can be written as
Therefore
Replacing
Integrating now gives the following solutions
Therefore
For solution (1) found earlier, since
Let
Solving for
This has the form
Where
Solving ode 1A
Taking derivative of (*) w.r.t.
Comparing the form
Hence (2) becomes
The singular solution is found by setting
Solving the above for
Substituting these in (1A) and keeping singular solution that verifies the ode gives
The general solution is found when
This ODE is now solved for
Unable to integrate (or intergal too complicated), and since no initial conditions are given, then the result can be written as
Substituing the above solution for
Solving ode 2A
Taking derivative of (*) w.r.t.
Comparing the form
Hence (2) becomes
The singular solution is found by setting
Solving the above for
Substituting these in (1A) and keeping singular solution that verifies the ode gives
The general solution is found when
This ODE is now solved for
Unable to integrate (or intergal too complicated), and since no initial conditions are given, then the result can be written as
Substituing the above solution for
Solving ode 3A
Taking derivative of (*) w.r.t.
Comparing the form
Hence (2) becomes
The singular solution is found by setting
Solving the above for
Substituting these in (1A) and keeping singular solution that verifies the ode gives
The general solution is found when
This ODE is now solved for
Unable to integrate (or intergal too complicated), and since no initial conditions are given, then the result can be written as
Substituing the above solution for
Solving ode 4A
Taking derivative of (*) w.r.t.
Comparing the form
Hence (2) becomes
The singular solution is found by setting
Solving the above for
Substituting these in (1A) and keeping singular solution that verifies the ode gives
The general solution is found when
This ODE is now solved for
Unable to integrate (or intergal too complicated), and since no initial conditions are given, then the result can be written as
Substituing the above solution for
Solving ode 5A
Taking derivative of (*) w.r.t.
Comparing the form
Hence (2) becomes
The singular solution is found by setting
Solving the above for
Substituting these in (1A) and keeping singular solution that verifies the ode gives
The general solution is found when
This ODE is now solved for
Unable to integrate (or intergal too complicated), and since no initial conditions are given, then the result can be written as
Substituing the above solution for
For solution (2) found earlier, since
Since the ode has the form
For solution (3) found earlier, since
Unable to integrate (or intergal too complicated), and since no initial conditions are given, then the result can be written as
Will add steps showing solving for IC soon.
The solution
was found not to satisfy the ode or the IC. Hence it is removed. The solution
was found not to satisfy the ode or the IC. Hence it is removed. The solution
was found not to satisfy the ode or the IC. Hence it is removed. The solution
was found not to satisfy the ode or the IC. Hence it is removed.
Summary of solutions found
ode:=y(x)*diff(diff(y(x),x),x)^3+y(x)^3*diff(y(x),x)^5 = 0; dsolve(ode,y(x), singsol=all);
Maple trace
`Methods for second order ODEs: *** Sublevel 2 *** Methods for second order ODEs: Successful isolation of d^2y/dx^2: 3 solutions were found. Trying to solve each resulting ODE. *** Sublevel 3 *** 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^2*_b(_a)^2)^(1/3)*_b(_a) = 0, _b(_a), HINT = [[_a, 5*_b]]` symmetry methods on request `, `1st order, trying reduction of order with given symmetries:`[_a, 5*_b]
ode=y[x]*D[y[x],{x,2}]^3+y[x]^3*D[y[x],x]^5==0; ic={}; DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
from sympy import * x = symbols("x") y = Function("y") ode = Eq(y(x)**3*Derivative(y(x), x)**5 + y(x)*Derivative(y(x), (x, 2))**3,0) ics = {} dsolve(ode,func=y(x),ics=ics)