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
:
Sunday, March 30, 2025 at 11:01:37 AM
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: 5.256 (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: 1.789 (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)
\[
\left [ y{\left (x \right )} = \sqrt [3]{2} \sqrt [3]{\frac {e^{- x}}{C_{1} e^{2 x} + 3}}, \ y{\left (x \right )} = \frac {\sqrt [3]{2} \sqrt [3]{\frac {e^{- x}}{C_{1} e^{2 x} + 3}} \left (-1 - \sqrt {3} i\right )}{2}, \ y{\left (x \right )} = \frac {\sqrt [3]{2} \sqrt [3]{\frac {e^{- x}}{C_{1} e^{2 x} + 3}} \left (-1 + \sqrt {3} i\right )}{2}\right ]
\]