67.7.2 problem Problem 3(b)
Internal
problem
ID
[14034]
Book
:
APPLIED
DIFFERENTIAL
EQUATIONS
The
Primary
Course
by
Vladimir
A.
Dobrushkin.
CRC
Press
2015
Section
:
Chapter
8.3
Systems
of
Linear
Differential
Equations
(Variation
of
Parameters).
Problems
page
514
Problem
number
:
Problem
3(b)
Date
solved
:
Wednesday, March 05, 2025 at 10:26:36 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=-7 x \left (t \right )+6 y \left (t \right )+6 \,{\mathrm e}^{-t}\\ \frac {d}{d t}y \left (t \right )&=-12 x \left (t \right )+5 y \left (t \right )+37 \end{align*}
✓ Maple. Time used: 1.092 (sec). Leaf size: 81
ode:=[diff(x(t),t) = -7*x(t)+6*y(t)+6*exp(-t), diff(y(t),t) = -12*x(t)+5*y(t)+37];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= 6+\frac {{\mathrm e}^{-t} \left (-2+\cos \left (6 t \right ) c_{1} -\cos \left (6 t \right ) c_{2} +\sin \left (6 t \right ) c_{1} +\sin \left (6 t \right ) c_{2} -2 \cos \left (6 t \right )-2 \sin \left (6 t \right )\right )}{2} \\
y &= 7+{\mathrm e}^{-t} \left (-2+\cos \left (6 t \right ) c_{1} +\sin \left (6 t \right ) c_{2} -2 \cos \left (6 t \right )\right ) \\
\end{align*}
✓ Mathematica. Time used: 0.123 (sec). Leaf size: 227
ode={D[x[t],t]==-7*x[t]+6*y[t]+6*Exp[-t],D[y[t],t]==-12*x[t]+5*y[t]+37};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to e^{-t} \left ((\cos (6 t)-\sin (6 t)) \int _1^t\left (6 \cos (6 K[1])-\left (-6+37 e^{K[1]}\right ) \sin (6 K[1])\right )dK[1]+\sin (6 t) \int _1^t\left (37 e^{K[2]} (\cos (6 K[2])-\sin (6 K[2]))+12 \sin (6 K[2])\right )dK[2]+c_1 \cos (6 t)-c_1 \sin (6 t)+c_2 \sin (6 t)\right ) \\
y(t)\to e^{-t} \left (-2 \sin (6 t) \int _1^t\left (6 \cos (6 K[1])-\left (-6+37 e^{K[1]}\right ) \sin (6 K[1])\right )dK[1]+(\sin (6 t)+\cos (6 t)) \int _1^t\left (37 e^{K[2]} (\cos (6 K[2])-\sin (6 K[2]))+12 \sin (6 K[2])\right )dK[2]+c_2 \cos (6 t)-2 c_1 \sin (6 t)+c_2 \sin (6 t)\right ) \\
\end{align*}
✓ Sympy. Time used: 0.304 (sec). Leaf size: 133
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(7*x(t) - 6*y(t) + Derivative(x(t), t) - 6*exp(-t),0),Eq(12*x(t) - 5*y(t) + Derivative(y(t), t) - 37,0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = - \left (\frac {C_{1}}{2} - \frac {C_{2}}{2}\right ) e^{- t} \sin {\left (6 t \right )} + \left (\frac {C_{1}}{2} + \frac {C_{2}}{2}\right ) e^{- t} \cos {\left (6 t \right )} + 6 \sin ^{2}{\left (6 t \right )} + 6 \cos ^{2}{\left (6 t \right )} - e^{- t} \sin ^{2}{\left (6 t \right )} - e^{- t} \cos ^{2}{\left (6 t \right )}, \ y{\left (t \right )} = - C_{1} e^{- t} \sin {\left (6 t \right )} + C_{2} e^{- t} \cos {\left (6 t \right )} + 7 \sin ^{2}{\left (6 t \right )} + 7 \cos ^{2}{\left (6 t \right )} - 2 e^{- t} \sin ^{2}{\left (6 t \right )} - 2 e^{- t} \cos ^{2}{\left (6 t \right )}\right ]
\]