23.4.197 problem 197

Internal problem ID [6499]
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 : 197
Date solved : Tuesday, September 30, 2025 at 03:02:22 PM
CAS classification : [[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 19

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

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

✓ Mathematica. Time used: 0.377 (sec). Leaf size: 17

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

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

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(a*y(x)*Derivative(y(x), (x, 2)) - (a - 1)*Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -sqrt(a*y(x)*Derivative(y(x), (x, 2))/(a - 1)) + Derivative(y(x)