23.4.25 problem 25

Internal problem ID [6327]
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 : 25
Date solved : Tuesday, September 30, 2025 at 02:47:26 PM
CAS classification : [NONE]

\begin{align*} y^{\prime \prime }&=\operatorname {g3} \left (x \right )+\operatorname {g2} \left (x \right ) y+\operatorname {g1} \left (x \right ) y^{2}+\operatorname {g0} \left (x \right ) y^{3}+\left (\operatorname {f1} \left (x \right )+\operatorname {f0} \left (x \right ) y\right ) y^{\prime } \end{align*}

✗ Maple

ode:=diff(diff(y(x),x),x) = g3(x)+g2(x)*y(x)+g1(x)*y(x)^2+g0(x)*y(x)^3+(f1(x)+f0(x)*y(x))*diff(y(x),x); 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✗ Mathematica

ode=D[y[x],{x,2}] == g3[x] + g2[x]*y[x] + g1[x]*y[x]^2 + g0[x]*y[x]^3 + (f1[x] + f0[x]*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(-(f0(x)*y(x) + f1(x))*Derivative(y(x), x) - g0(x)*y(x)**3 - g1(x)*y(x)**2 - g2(x)*y(x) - g3(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (-g0(x)*y(x)**3 - g1(x)*y(x)**2 - g2(x)*y(