82.8.21 problem 36-22

Internal problem ID [21892]
Book : The Differential Equations Problem Solver. VOL. II. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 36. Nonlinear differential equations. Page 1203
Problem number : 36-22
Date solved : Thursday, October 02, 2025 at 08:05:48 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 29

ode:=(diff(y(x),x)^2-y(x)^2)*exp(diff(y(x),x))-x*diff(y(x),x)^2+x*y(x)^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= c_1 \,{\mathrm e}^{x} \\ y &= c_1 \,{\mathrm e}^{-x} \\ y &= x \ln \left (x \right )-x +c_1 \\ \end{align*}

✓ Mathematica. Time used: 0.029 (sec). Leaf size: 42

ode=(D[y[x],x]^2-y[x]^2)*Exp[ D[y[x],x] ]-x*D[y[x],x]^2+x*y[x]^2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_1 e^{-x}\\ y(x)&\to c_1 e^x\\ y(x)&\to -x+x \log (x)+c_1\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 0.144 (sec). Leaf size: 24

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

\[ \left [ y{\left (x \right )} = C_{1} + x \log {\left (x \right )} - x, \ y{\left (x \right )} = C_{1} e^{x}, \ y{\left (x \right )} = C_{1} e^{- x}\right ] \]