82.8.7 problem 36-8

Internal problem ID [21878]
Book : The Differential Equations Problem Solver. VOL. II. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 36. Nonlinear differential equations. Page 1203
Problem number : 36-8
Date solved : Thursday, October 02, 2025 at 08:03:16 PM
CAS classification : [[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 17

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

\begin{align*} y &= 0 \\ y &= \frac {\left (c_1 x +c_2 \right )^{2}}{4} \\ \end{align*}

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 29

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

\begin{align*} y(x)&\to \frac {(c_1 x+2 c_2){}^2}{4 c_2}\\ y(x)&\to \text {Indeterminate} \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE -sqrt(2)*sqrt(y(x)*Derivative(y(x), (x, 2))) + Derivative(y(x), x) cannot be solved by the factorable group method