23.4.147 problem 147

Internal problem ID [6449]
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 : 147
Date solved : Tuesday, September 30, 2025 at 02:57:09 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _with_potential_symmetries], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.014 (sec). Leaf size: 21

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

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

✓ Mathematica. Time used: 0.744 (sec). Leaf size: 43

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

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

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-y(x)**2*Derivative(y(x), x) + 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((y(x)**3 + 4*Derivative(y(x), (x, 2)))*y(x))/2 + y(x)**2/2