23.4.123 problem 123

Internal problem ID [6425]
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 : 123
Date solved : Friday, October 03, 2025 at 02:05:42 AM
CAS classification : [NONE]

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

✓ Maple. Time used: 0.068 (sec). Leaf size: 121

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

\begin{align*} y &= 0 \\ y &= \sqrt {f \left (x \right )} \\ y &= -\sqrt {f \left (x \right )} \\ y &= \operatorname {RootOf}\left (-63 \int \sqrt {f \left (x \right )}d x -\int _{}^{\textit {\_Z}}\frac {4 {\operatorname {RootOf}\left (\left (-4 \textit {\_f}^{6}+12 \textit {\_f}^{4}-12 \textit {\_f}^{2}+320 c_1 +4\right ) \textit {\_Z}^{9}+\left (-189 \textit {\_f}^{6}+567 \textit {\_f}^{4}-567 \textit {\_f}^{2}+15120 c_1 +189\right ) \textit {\_Z}^{6}+238140 c_1 \,\textit {\_Z}^{3}+1250235 c_1 \right )}^{3}+63}{\textit {\_f}^{2}-1}d \textit {\_f} +63 c_2 \right ) \sqrt {f \left (x \right )} \\ \end{align*}

✗ Mathematica

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

Not solved

✗ Sympy

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

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