32.2.22 problem 22

Internal problem ID [7739]
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 : Tuesday, September 30, 2025 at 05:04:14 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _Bernoulli]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 136

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

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

✓ Mathematica. Time used: 4.608 (sec). Leaf size: 90

ode=D[y[x],x]+y[x]==y[x]^4*Exp[x]; 
ic={}; 
DSolve[{ode,ic},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*}

✓ Sympy. Time used: 1.219 (sec). Leaf size: 94

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-y(x)**4*exp(x) + y(x) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = \sqrt [3]{2} \sqrt [3]{\frac {e^{- x}}{C_{1} e^{2 x} + 3}}, \ y{\left (x \right )} = \frac {\sqrt [3]{2} \sqrt [3]{\frac {e^{- x}}{C_{1} e^{2 x} + 3}} \left (-1 - \sqrt {3} i\right )}{2}, \ y{\left (x \right )} = \frac {\sqrt [3]{2} \sqrt [3]{\frac {e^{- x}}{C_{1} e^{2 x} + 3}} \left (-1 + \sqrt {3} i\right )}{2}\right ] \]