34.15.3 problem 12
Internal
problem
ID
[8075]
Book
:
Schaums
Outline.
Theory
and
problems
of
Differential
Equations,
1st
edition.
Frank
Ayres.
McGraw
Hill
1952
Section
:
Chapter
21.
System
of
simultaneous
linear
equations.
Supplemetary
problems.
Page
163
Problem
number
:
12
Date
solved
:
Tuesday, September 30, 2025 at 05:15:14 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )+x \left (t \right )+2 \frac {d}{d t}y \left (t \right )+7 y \left (t \right )&={\mathrm e}^{t}+2\\ -2 x \left (t \right )+\frac {d}{d t}y \left (t \right )+3 y \left (t \right )&={\mathrm e}^{t}-1 \end{align*}
✓ Maple. Time used: 0.263 (sec). Leaf size: 70
ode:=[diff(x(t),t)+x(t)+2*diff(y(t),t)+7*y(t) = exp(t)+2, -2*x(t)+diff(y(t),t)+3*y(t) = exp(t)-1];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= {\mathrm e}^{-4 t} \sin \left (t \right ) c_2 +{\mathrm e}^{-4 t} \cos \left (t \right ) c_1 +\frac {13}{17}-\frac {5 \,{\mathrm e}^{t}}{26} \\
y \left (t \right ) &= -{\mathrm e}^{-4 t} \sin \left (t \right ) c_2 -{\mathrm e}^{-4 t} \cos \left (t \right ) c_2 -{\mathrm e}^{-4 t} \cos \left (t \right ) c_1 +{\mathrm e}^{-4 t} \sin \left (t \right ) c_1 +\frac {2 \,{\mathrm e}^{t}}{13}+\frac {3}{17} \\
\end{align*}
✓ Mathematica. Time used: 0.102 (sec). Leaf size: 209
ode={D[x[t],t]+x[t]+2*D[y[t],t]+7*y[t]==Exp[t]+2,-2*x[t]+D[y[t],t]+3*y[t]==Exp[t]-1};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*} x(t)&\to e^{-4 t} \left ((\cos (t)-\sin (t)) \int _1^te^{4 K[1]} \left (3 \sin (K[1])-\left (-4+e^{K[1]}\right ) \cos (K[1])\right )dK[1]-\sin (t) \int _1^te^{4 K[2]} \left (\left (-1+e^{K[2]}\right ) \cos (K[2])+\left (-7+e^{K[2]}\right ) \sin (K[2])\right )dK[2]+c_1 \cos (t)-c_1 \sin (t)-c_2 \sin (t)\right )\\ y(t)&\to e^{-4 t} \left (2 \sin (t) \int _1^te^{4 K[1]} \left (3 \sin (K[1])-\left (-4+e^{K[1]}\right ) \cos (K[1])\right )dK[1]+(\sin (t)+\cos (t)) \int _1^te^{4 K[2]} \left (\left (-1+e^{K[2]}\right ) \cos (K[2])+\left (-7+e^{K[2]}\right ) \sin (K[2])\right )dK[2]+c_2 \cos (t)+2 c_1 \sin (t)+c_2 \sin (t)\right ) \end{align*}
✓ Sympy. Time used: 2.868 (sec). Leaf size: 136
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(x(t) + 7*y(t) - exp(t) + Derivative(x(t), t) + 2*Derivative(y(t), t) - 2,0),Eq(-2*x(t) + 3*y(t) - exp(t) + Derivative(y(t), t) + 1,0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = - \left (\frac {C_{1}}{2} - \frac {C_{2}}{2}\right ) e^{- 4 t} \sin {\left (t \right )} - \left (\frac {C_{1}}{2} + \frac {C_{2}}{2}\right ) e^{- 4 t} \cos {\left (t \right )} - \frac {5 e^{t} \sin ^{2}{\left (t \right )}}{26} - \frac {5 e^{t} \cos ^{2}{\left (t \right )}}{26} + \frac {13 \sin ^{2}{\left (t \right )}}{17} + \frac {13 \cos ^{2}{\left (t \right )}}{17}, \ y{\left (t \right )} = C_{1} e^{- 4 t} \cos {\left (t \right )} - C_{2} e^{- 4 t} \sin {\left (t \right )} + \frac {2 e^{t} \sin ^{2}{\left (t \right )}}{13} + \frac {2 e^{t} \cos ^{2}{\left (t \right )}}{13} + \frac {3 \sin ^{2}{\left (t \right )}}{17} + \frac {3 \cos ^{2}{\left (t \right )}}{17}\right ]
\]