64.6.6 problem 6

Internal problem ID [13285]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 2, Miscellaneous Review. Exercises page 60
Problem number : 6
Date solved : Monday, March 31, 2025 at 07:45:30 AM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 58

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

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

✓ Mathematica. Time used: 0.092 (sec). Leaf size: 29

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

\begin{align*} y(x)\to 0 \\ y(x)\to \frac {c_1}{\sqrt {e^{2 x}-2}} \\ y(x)\to 0 \\ \end{align*}

✓ Sympy. Time used: 0.348 (sec). Leaf size: 14

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

\[ y{\left (x \right )} = \frac {C_{1}}{\sqrt {e^{2 x} - 2}} \]