23.4.293 problem 296

Internal problem ID [6595]
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 : 296
Date solved : Tuesday, September 30, 2025 at 03:27:18 PM
CAS classification : [[_2nd_order, _missing_y]]

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

✓ Maple. Time used: 0.162 (sec). Leaf size: 44

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

\begin{align*} y &= \frac {\sqrt {a}\, x^{2}}{2}+c_{1} \\ y &= -\frac {\sqrt {a}\, x^{2}}{2}+c_{1} \\ y &= c_{2} x^{3}+\frac {a x}{12 c_{2}}+c_{1} \\ \end{align*}

✗ Mathematica

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

{}

✗ Sympy

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

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