23.4.122 problem 122

Internal problem ID [6424]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 4. THE NONLINEAR EQUATION OF SECOND ORDER, page 380
Problem number : 122
Date solved : Tuesday, September 30, 2025 at 02:56:32 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} f \left (x \right )^{2} y^{\prime \prime }&=3 f \left (x \right )^{3}-a f \left (x \right )^{5}-f \left (x \right )^{2} y+3 f \left (x \right ) f^{\prime }\left (x \right ) \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 76

ode:=f(x)^2*diff(diff(y(x),x),x) = 3*f(x)^3-a*f(x)^5-f(x)^2*y(x)+3*f(x)*diff(f(x),x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \sin \left (x \right ) c_2 +\cos \left (x \right ) c_1 -\int \frac {\cos \left (x \right ) \left (f \left (x \right )^{4} a -3 f \left (x \right )^{2}-3 f^{\prime }\left (x \right )\right )}{f \left (x \right )}d x \sin \left (x \right )+\int \frac {\sin \left (x \right ) \left (f \left (x \right )^{4} a -3 f \left (x \right )^{2}-3 f^{\prime }\left (x \right )\right )}{f \left (x \right )}d x \cos \left (x \right ) \]

✓ Mathematica. Time used: 0.785 (sec). Leaf size: 99

ode=f[x]^2*D[y[x],{x,2}] == 3*f[x]^3 - a*f[x]^5 - f[x]^2*y[x] + 3*f[x]*D[f[x],x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \cos (x) \int _1^x\frac {\sin (K[1]) \left (a f(K[1])^4-3 f(K[1])^2-3 f'(K[1])\right )}{f(K[1])}dK[1]+\sin (x) \int _1^x-\frac {\cos (K[2]) \left (a f(K[2])^4-3 f(K[2])^2-3 f'(K[2])\right )}{f(K[2])}dK[2]+c_1 \cos (x)+c_2 \sin (x) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(a*f(x)**5 - 3*f(x)**3 + f(x)**2*y(x) + f(x)**2*Derivative(y(x), (x, 2)) - 3*f(x)*Derivative(f(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE a*f(x)**4 - 3*f(x)**2 + f(x)*y(x) + f(x)*Derivative(y(x), (x, 2)