38.2.22 problem 22

Internal problem ID [6451]
Book : Engineering Mathematics. By K. A. Stroud. 5th edition. Industrial press Inc. NY. 2001
Section : Program 24. First order differential equations. Further problems 24. page 1068
Problem number : 22
Date solved : Monday, January 27, 2025 at 02:04:33 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _Bernoulli]

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

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 136

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

\begin{align*} y &= \frac {2^{{1}/{3}} \left ({\mathrm e}^{2 x} \left (2 \,{\mathrm e}^{2 x} c_1 +3\right )^{2}\right )^{{1}/{3}} {\mathrm e}^{-x}}{2 \,{\mathrm e}^{2 x} c_1 +3} \\ y &= -\frac {\left (1+i \sqrt {3}\right ) 2^{{1}/{3}} \left ({\mathrm e}^{2 x} \left (2 \,{\mathrm e}^{2 x} c_1 +3\right )^{2}\right )^{{1}/{3}} {\mathrm e}^{-x}}{4 \,{\mathrm e}^{2 x} c_1 +6} \\ y &= \frac {2^{{1}/{3}} \left ({\mathrm e}^{2 x} \left (2 \,{\mathrm e}^{2 x} c_1 +3\right )^{2}\right )^{{1}/{3}} \left (i \sqrt {3}-1\right ) {\mathrm e}^{-x}}{4 \,{\mathrm e}^{2 x} c_1 +6} \\ \end{align*}

✓ Solution by Mathematica

Time used: 5.256 (sec). Leaf size: 90

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

\begin{align*} y(x)\to -\frac {\sqrt [3]{-2}}{\sqrt [3]{e^x \left (3+2 c_1 e^{2 x}\right )}} \\ y(x)\to \frac {1}{\sqrt [3]{\frac {3 e^x}{2}+c_1 e^{3 x}}} \\ y(x)\to \frac {(-1)^{2/3}}{\sqrt [3]{\frac {3 e^x}{2}+c_1 e^{3 x}}} \\ y(x)\to 0 \\ \end{align*}