6.3.30 problem 38

Internal problem ID [1607]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 2, First order equations. separable equations. Section 2.2 Page 52
Problem number : 38
Date solved : Tuesday, September 30, 2025 at 04:39:48 AM
CAS classification : [[_Abel, `2nd type`, `class A`]]

\begin{align*} y^{\prime }-2 y&=\frac {x \,{\mathrm e}^{2 x}}{1-y \,{\mathrm e}^{-2 x}} \end{align*}

✓ Maple. Time used: 0.010 (sec). Leaf size: 60

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

\begin{align*} y &= {\mathrm e}^{2 x}+{\mathrm e}^{4 x} \sqrt {-{\mathrm e}^{-4 x} \left (x^{2}+2 c_1 -1\right )} \\ y &= {\mathrm e}^{2 x}-{\mathrm e}^{4 x} \sqrt {-{\mathrm e}^{-4 x} \left (x^{2}+2 c_1 -1\right )} \\ \end{align*}

✓ Mathematica. Time used: 0.466 (sec). Leaf size: 72

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

\begin{align*} y(x)&\to e^{2 x}-\frac {\sqrt {x^2-1-c_1}}{\sqrt {-e^{-4 x}}}\\ y(x)&\to e^{2 x}+\frac {\sqrt {x^2-1-c_1}}{\sqrt {-e^{-4 x}}} \end{align*}

✗ Sympy

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

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