23.4.298 problem 301

Internal problem ID [6600]
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 : 301
Date solved : Friday, October 03, 2025 at 02:09:36 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} h y^{2}+\operatorname {g1} y y^{\prime }+\operatorname {g0} {y^{\prime }}^{2}+\operatorname {f2} y y^{\prime \prime }+\operatorname {f1} y^{\prime } y^{\prime \prime }+\operatorname {f0} {y^{\prime \prime }}^{2}&=0 \end{align*}

✓ Maple. Time used: 1.923 (sec). Leaf size: 141

ode:=h*y(x)^2+g1*y(x)*diff(y(x),x)+g0*diff(y(x),x)^2+f2*y(x)*diff(diff(y(x),x),x)+f1*diff(y(x),x)*diff(diff(y(x),x),x)+f0*diff(diff(y(x),x),x)^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= 0 \\ y &= {\mathrm e}^{\int \operatorname {RootOf}\left (x -2 \operatorname {f0} \int _{}^{\textit {\_Z}}-\frac {1}{2 \textit {\_f}^{2} \operatorname {f0} +\textit {\_f} \operatorname {f1} -\sqrt {-4 \textit {\_f}^{2} \operatorname {f0} \operatorname {g0} +\textit {\_f}^{2} \operatorname {f1}^{2}-4 \textit {\_f} \operatorname {f0} \operatorname {g1} +2 \textit {\_f} \operatorname {f1} \operatorname {f2} -4 \operatorname {f0} h +\operatorname {f2}^{2}}+\operatorname {f2}}d \textit {\_f} +c_1 \right )d x +c_2} \\ y &= {\mathrm e}^{\int \operatorname {RootOf}\left (x +2 \operatorname {f0} \int _{}^{\textit {\_Z}}\frac {1}{2 \textit {\_f}^{2} \operatorname {f0} +\textit {\_f} \operatorname {f1} +\sqrt {-4 \textit {\_f}^{2} \operatorname {f0} \operatorname {g0} +\textit {\_f}^{2} \operatorname {f1}^{2}-4 \textit {\_f} \operatorname {f0} \operatorname {g1} +2 \textit {\_f} \operatorname {f1} \operatorname {f2} -4 \operatorname {f0} h +\operatorname {f2}^{2}}+\operatorname {f2}}d \textit {\_f} +c_1 \right )d x +c_2} \\ \end{align*}

✗ Mathematica

ode=h*y[x]^2 + g1*y[x]*D[y[x],x] + g0*D[y[x],x]^2 + f2*y[x]*D[y[x],{x,2}] + f1*D[y[x],x]*D[y[x],{x,2}] + f0*D[y[x],{x,2}]^2 == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
f0 = symbols("f0") 
f1 = symbols("f1") 
f2 = symbols("f2") 
g0 = symbols("g0") 
g1 = symbols("g1") 
h = symbols("h") 
y = Function("y") 
ode = Eq(f0*Derivative(y(x), (x, 2))**2 + f1*Derivative(y(x), x)*Derivative(y(x), (x, 2)) + f2*y(x)*Derivative(y(x), (x, 2)) + g0*Derivative(y(x), x)**2 + g1*y(x)*Derivative(y(x), x) + h*y(x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (-f1*Derivative(y(x), (x, 2)) - g1*y(x) +