Internal problem ID [956]
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.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_symmetry_[F(x),G(x)*y+H(x)]]]
Solve \begin {gather*} \boxed {y^{\prime }-y-\frac {\left (x +1\right ) {\mathrm e}^{4 x}}{\left (y+{\mathrm e}^{x}\right )^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 112
dsolve(diff(y(x),x)-y(x)=((x+1)*exp(4*x))/(y(x)+exp(x))^2,y(x), singsol=all)
\begin{align*} y \relax (x ) = \left (3 x \,{\mathrm e}^{x}-3 c_{1}+1\right )^{\frac {1}{3}} {\mathrm e}^{x}-{\mathrm e}^{x} \\ y \relax (x ) = \left (-\frac {\left (3 x \,{\mathrm e}^{x}-3 c_{1}+1\right )^{\frac {1}{3}}}{2}-\frac {i \sqrt {3}\, \left (3 x \,{\mathrm e}^{x}-3 c_{1}+1\right )^{\frac {1}{3}}}{2}\right ) {\mathrm e}^{x}-{\mathrm e}^{x} \\ y \relax (x ) = \left (-\frac {\left (3 x \,{\mathrm e}^{x}-3 c_{1}+1\right )^{\frac {1}{3}}}{2}+\frac {i \sqrt {3}\, \left (3 x \,{\mathrm e}^{x}-3 c_{1}+1\right )^{\frac {1}{3}}}{2}\right ) {\mathrm e}^{x}-{\mathrm e}^{x} \\ \end{align*}
✓ Solution by Mathematica
Time used: 19.127 (sec). Leaf size: 143
DSolve[y'[x]-y[x]==((x+1)*Exp[4*x])/(y[x]+Exp[x])^2,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*}