23.4.143 problem 143

Internal problem ID [6445]
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 : 143
Date solved : Tuesday, September 30, 2025 at 02:56:56 PM
CAS classification : [NONE]

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

✗ Maple

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

\[ \text {No solution found} \]

✓ Mathematica. Time used: 24.817 (sec). Leaf size: 197

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

\begin{align*} y(x)&\to -\frac {\exp \left (c_2-\int _1^x\frac {y(K[3])^3+\left (c_1+\int _1^{K[3]}-\frac {y(K[1])^3-f'(K[1]) y(K[1])+f(K[1]) y'(K[1])}{y(K[1])^2}dK[1]\right ){}^2 y(K[3])^2-f'(K[3]) y(K[3])+f(K[3]) y'(K[3])}{y(K[3])^2 \left (c_1+\int _1^{K[3]}-\frac {y(K[1])^3-f'(K[1]) y(K[1])+f(K[1]) y'(K[1])}{y(K[1])^2}dK[1]\right )}dK[3]\right )}{\int _1^x-\frac {y(K[1])^3-f'(K[1]) y(K[1])+f(K[1]) y'(K[1])}{y(K[1])^2}dK[1]+c_1}\\ y(x)&\to 0 \end{align*}

✗ Sympy

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

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