15.18.25 problem 25

Internal problem ID [3268]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 35, page 157
Problem number : 25
Date solved : Tuesday, March 04, 2025 at 04:11:23 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}&=0 \end{align*}

✓ Maple. Time used: 0.019 (sec). Leaf size: 17

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

\begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= \frac {\left (c_{1} x +c_{2} \right )^{2}}{4} \\ \end{align*}

✓ Mathematica. Time used: 0.019 (sec). Leaf size: 29

ode=2*y[x]*D[y[x],{x,2}]-D[y[x],x]^2==0; 
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