12.5.48 problem 47

Internal problem ID [1672]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 2, First order equations. Transformation of Nonlinear Equations into Separable Equations. Section 2.4 Page 68
Problem number : 47
Date solved : Monday, January 27, 2025 at 05:24:07 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _Riccati]

\begin{align*} y^{\prime }&=y^{2} {\mathrm e}^{-x}+4 y+2 \,{\mathrm e}^{x} \end{align*}

✓ Solution by Maple

Time used: 0.018 (sec). Leaf size: 22

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

\[ y = -\frac {2 \,{\mathrm e}^{x} \left ({\mathrm e}^{x} c_1 -1\right )}{-2+{\mathrm e}^{x} c_1} \]

✓ Solution by Mathematica

Time used: 0.280 (sec). Leaf size: 30

DSolve[D[y[x],x]==y[x]^2*Exp[-x]+4*y[x]+2*Exp[x],y[x],x,IncludeSingularSolutions -> True]
 

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