23.4.151 problem 151

Internal problem ID [6453]
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 : 151
Date solved : Tuesday, September 30, 2025 at 02:57:18 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 34

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

\begin{align*} y &= 0 \\ y &= \frac {2 \,{\mathrm e}^{\sqrt {2}\, x} \sqrt {2}}{{\mathrm e}^{2 \sqrt {2}\, x} c_1 -c_2} \\ \end{align*}

✓ Mathematica. Time used: 1.002 (sec). Leaf size: 47

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

\begin{align*} y(x)&\to \frac {c_2 \sqrt {e^{2 \sqrt {2} (x+c_1)}}}{1+e^{2 \sqrt {2} (x+c_1)}}\\ y(x)&\to 0 \end{align*}

✗ Sympy

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

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