23.4.140 problem 140

Internal problem ID [6442]
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 : 140
Date solved : Tuesday, September 30, 2025 at 02:56:54 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.018 (sec). Leaf size: 20

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

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

✓ Mathematica. Time used: 0.654 (sec). Leaf size: 26

ode=y[x]*D[y[x],{x,2}] == -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 {-2+e^{c_1 (x+c_2)}}{c_1}\\ y(x)&\to \text {Indeterminate} \end{align*}

✗ Sympy

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

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