23.4.153 problem 153

Internal problem ID [6455]
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 : 153
Date solved : Tuesday, September 30, 2025 at 02:57:27 PM
CAS classification : [[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 28

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

\begin{align*} y &= 0 \\ y &= \left (-\frac {1}{\left (a -1\right ) \left (c_1 x +c_2 \right )}\right )^{\frac {1}{a -1}} \\ \end{align*}

✓ Mathematica. Time used: 0.271 (sec). Leaf size: 26

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

\begin{align*} y(x)&\to c_2 (-a x+x-c_1){}^{\frac {1}{1-a}} \end{align*}

✗ Sympy

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

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