69.8.2 problem 200

Internal problem ID [18105]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 8. First order not solved for the derivative. Exercises page 67
Problem number : 200
Date solved : Thursday, October 02, 2025 at 02:42:23 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.053 (sec). Leaf size: 27

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

\begin{align*} y &= 0 \\ y &= c_1 \,{\mathrm e}^{x -{\mathrm e}^{x}} \\ y &= c_1 \,{\mathrm e}^{x +{\mathrm e}^{x}} \\ \end{align*}

✓ Mathematica. Time used: 0.053 (sec). Leaf size: 36

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

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

✓ Sympy. Time used: 0.264 (sec). Leaf size: 20

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

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