23.4.237 problem 237

Internal problem ID [6539]
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 : 237
Date solved : Friday, October 03, 2025 at 02:09:28 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} a x +y {y^{\prime }}^{2}+y^{2} y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.033 (sec). Leaf size: 153

ode:=a*x+y(x)*diff(y(x),x)^2+y(x)^2*diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \operatorname {RootOf}\left (-\ln \left (x \right )+\int _{}^{\textit {\_Z}}-\frac {\textit {\_f}^{2}-\operatorname {RootOf}\left (3 \ln \left (\textit {\_f}^{2}-\textit {\_f} \,a^{{1}/{3}}+a^{{2}/{3}}\right )-18 c_1 -6 \ln \left (\textit {\_f} +a^{{1}/{3}}\right )-6 \sqrt {3}\, \arctan \left (\frac {\left (\textit {\_f} \,a^{{1}/{3}}-2 \textit {\_Z} \right ) \sqrt {3}}{3 \textit {\_f} \,a^{{1}/{3}}}\right )+6 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (-2 \textit {\_f} +a^{{1}/{3}}\right )}{3 a^{{1}/{3}}}\right )+6 \ln \left (\frac {\textit {\_f} \,a^{{1}/{3}}+\textit {\_Z}}{\textit {\_f} \,a^{{1}/{3}}}\right )-3 \ln \left (\frac {\textit {\_f}^{2} a^{{2}/{3}}-\textit {\_Z} \,a^{{1}/{3}} \textit {\_f} +\textit {\_Z}^{2}}{\textit {\_f}^{2} a^{{2}/{3}}}\right )\right )}{\textit {\_f}^{3}+a}d \textit {\_f} +c_2 \right ) x \]

✗ Mathematica

ode=a*x + y[x]*D[y[x],x]^2 + y[x]^2*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

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

NotImplementedError : The given ODE -sqrt(-a*x/y(x) - y(x)*Derivative(y(x), (x, 2))) + Derivative(y(