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 : Monday, January 27, 2025 at 05:13:43 AM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

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

✓ Solution by Maple

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

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

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

✓ Solution by Mathematica

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

DSolve[D[y[x],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*}