12.3.29 problem 37

Internal problem ID [1606]
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 : 37
Date solved : Saturday, March 29, 2025 at 11:07:13 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

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

✓ Maple. Time used: 0.013 (sec). Leaf size: 90

ode:=diff(y(x),x)-y(x) = (1+x)*exp(4*x)/(y(x)+exp(x))^2; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= {\mathrm e}^{x} \left (\left (3 x \,{\mathrm e}^{x}-3 c_1 +1\right )^{{1}/{3}}-1\right ) \\ y &= -\frac {{\mathrm e}^{x} \left (i \sqrt {3}\, \left (3 x \,{\mathrm e}^{x}-3 c_1 +1\right )^{{1}/{3}}+\left (3 x \,{\mathrm e}^{x}-3 c_1 +1\right )^{{1}/{3}}+2\right )}{2} \\ y &= \frac {{\mathrm e}^{x} \left (i \sqrt {3}-1\right ) \left (3 x \,{\mathrm e}^{x}-3 c_1 +1\right )^{{1}/{3}}}{2}-{\mathrm e}^{x} \\ \end{align*}

✓ Mathematica. Time used: 19.352 (sec). Leaf size: 143

ode=D[y[x],x]-y[x]==((x+1)*Exp[4*x])/(y[x]+Exp[x])^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -e^x+e^{3 x} \sqrt [3]{e^{-6 x} \left (3 e^x x+1+3 c_1\right )} \\ y(x)\to -e^x+\frac {1}{2} i \left (\sqrt {3}+i\right ) e^{3 x} \sqrt [3]{e^{-6 x} \left (3 e^x x+1+3 c_1\right )} \\ y(x)\to -e^x-\frac {1}{2} \left (1+i \sqrt {3}\right ) e^{3 x} \sqrt [3]{e^{-6 x} \left (3 e^x x+1+3 c_1\right )} \\ \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((-x - 1)*exp(4*x)/(y(x) + exp(x))**2 - y(x) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out