2.22 problem 22

Internal problem ID [4600]

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.
ODE order: 1.
ODE degree: 1.

CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _Bernoulli]

Solve \begin {gather*} \boxed {y^{\prime }+y-y^{4} {\mathrm e}^{x}=0} \end {gather*}

Solution by Maple

Time used: 0.015 (sec). Leaf size: 194

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

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

Solution by Mathematica

Time used: 5.677 (sec). Leaf size: 90

DSolve[y'[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*}