23.4.295 problem 298

Internal problem ID [6597]
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 : 298
Date solved : Tuesday, September 30, 2025 at 03:27:20 PM
CAS classification : [NONE]

\begin{align*} 2 \left (x -y^{\prime }\right ) y^{\prime }-x \left (x +4 y^{\prime }\right ) y^{\prime \prime }+2 \left (x^{2}+1\right ) {y^{\prime \prime }}^{2}&=2 y \end{align*}

✗ Maple

ode:=2*(x-diff(y(x),x))*diff(y(x),x)-x*(x+4*diff(y(x),x))*diff(diff(y(x),x),x)+2*(x^2+1)*diff(diff(y(x),x),x)^2 = 2*y(x); 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✗ Mathematica

ode=2*(x - D[y[x],x])*D[y[x],x] - x*(x + 4*D[y[x],x])*D[y[x],{x,2}] + 2*(1 + x^2)*D[y[x],{x,2}]^2 == 2*y[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*(x + 4*Derivative(y(x), x))*Derivative(y(x), (x, 2)) + (2*x - 2*Derivative(y(x), x))*Derivative(y(x), x) + (2*x**2 + 2)*Derivative(y(x), (x, 2))**2 - 2*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE x*Derivative(y(x), (x, 2)) - x/2 + sqrt(8*x**2*Derivative(y(x),