80.7.23 problem B 15
Internal
problem
ID
[21342]
Book
:
A
Textbook
on
Ordinary
Differential
Equations
by
Shair
Ahmad
and
Antonio
Ambrosetti.
Second
edition.
ISBN
978-3-319-16407-6.
Springer
2015
Section
:
Chapter
7.
System
of
first
order
equations.
Excercise
7.6
at
page
162
Problem
number
:
B
15
Date
solved
:
Thursday, October 02, 2025 at 07:28:36 PM
CAS
classification
:
system_of_ODEs
\begin{align*} x^{\prime }&=x+2 y \left (t \right )+{\mathrm e}^{t}\\ y^{\prime }\left (t \right )&=x-2 y \left (t \right )-{\mathrm e}^{t} \end{align*}
✓ Maple. Time used: 0.062 (sec). Leaf size: 93
ode:=[diff(x(t),t) = x(t)+2*y(t)+exp(t), diff(y(t),t) = x(t)-2*y(t)-exp(t)];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= {\mathrm e}^{\frac {\left (-1+\sqrt {17}\right ) t}{2}} c_2 +{\mathrm e}^{-\frac {\left (1+\sqrt {17}\right ) t}{2}} c_1 -\frac {{\mathrm e}^{t}}{2} \\
y \left (t \right ) &= \frac {{\mathrm e}^{\frac {\left (-1+\sqrt {17}\right ) t}{2}} c_2 \sqrt {17}}{4}-\frac {{\mathrm e}^{-\frac {\left (1+\sqrt {17}\right ) t}{2}} c_1 \sqrt {17}}{4}-\frac {3 \,{\mathrm e}^{\frac {\left (-1+\sqrt {17}\right ) t}{2}} c_2}{4}-\frac {3 \,{\mathrm e}^{-\frac {\left (1+\sqrt {17}\right ) t}{2}} c_1}{4}-\frac {{\mathrm e}^{t}}{2} \\
\end{align*}
✓ Mathematica. Time used: 0.483 (sec). Leaf size: 184
ode={D[x[t],t]==x[t]+2*y[t]+Exp[t],D[y[t],t]==x[t]-2*y[t]-Exp[t]};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*} x(t)&\to \frac {1}{34} e^{-\frac {1}{2} \left (1+\sqrt {17}\right ) t} \left (-17 e^{\frac {1}{2} \left (3+\sqrt {17}\right ) t}+\left (\left (17+3 \sqrt {17}\right ) c_1+4 \sqrt {17} c_2\right ) e^{\sqrt {17} t}+\left (17-3 \sqrt {17}\right ) c_1-4 \sqrt {17} c_2\right )\\ y(t)&\to \frac {1}{34} e^{-\frac {1}{2} \left (1+\sqrt {17}\right ) t} \left (-17 e^{\frac {1}{2} \left (3+\sqrt {17}\right ) t}+\left (2 \sqrt {17} c_1+\left (17-3 \sqrt {17}\right ) c_2\right ) e^{\sqrt {17} t}-2 \sqrt {17} c_1+\left (17+3 \sqrt {17}\right ) c_2\right ) \end{align*}
✓ Sympy. Time used: 0.433 (sec). Leaf size: 85
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(-x(t) - 2*y(t) - exp(t) + Derivative(x(t), t),0),Eq(-x(t) + 2*y(t) + exp(t) + Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = \frac {C_{1} \left (3 + \sqrt {17}\right ) e^{- \frac {t \left (1 - \sqrt {17}\right )}{2}}}{2} + \frac {C_{2} \left (3 - \sqrt {17}\right ) e^{- \frac {t \left (1 + \sqrt {17}\right )}{2}}}{2} - \frac {e^{t}}{2}, \ y{\left (t \right )} = C_{1} e^{- \frac {t \left (1 - \sqrt {17}\right )}{2}} + C_{2} e^{- \frac {t \left (1 + \sqrt {17}\right )}{2}} - \frac {e^{t}}{2}\right ]
\]