66.2.44 problem Problem 59

Internal problem ID [13872]
Book : Differential equations and the calculus of variations by L. ElSGOLTS. MIR PUBLISHERS, MOSCOW, Third printing 1977.
Section : Chapter 2, DIFFERENTIAL EQUATIONS OF THE SECOND ORDER AND HIGHER. Problems page 172
Problem number : Problem 59
Date solved : Monday, March 31, 2025 at 08:15:49 AM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.033 (sec). Leaf size: 20

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

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

✓ Mathematica. Time used: 1.469 (sec). Leaf size: 26

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

\begin{align*} y(x)\to \frac {1+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 - Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -sqrt(4*y(x)*Derivative(y(x), (x, 2)) + 1)/2 + Derivative(y(x), x) + 1/2 cannot be solved by the factorable group method