23.4.32 problem 32

Internal problem ID [6334]
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 : 32
Date solved : Friday, October 03, 2025 at 02:01:33 AM
CAS classification : [[_2nd_order, _with_potential_symmetries]]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 34

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

\[ y = \frac {c_1 \int {\mathrm e}^{\int f \left (x \right )d x}d x +c_2}{c_1 \int \int {\mathrm e}^{\int f \left (x \right )d x}d x d x +c_2 x +1} \]

✗ Mathematica

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

Not solved

✗ Sympy

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

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