67.7.3 problem Problem 3(c)
Internal
problem
ID
[14035]
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(c)
Date
solved
:
Wednesday, March 05, 2025 at 10:26:37 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=-7 x \left (t \right )+10 y \left (t \right )+18 \,{\mathrm e}^{t}\\ \frac {d}{d t}y \left (t \right )&=-10 x \left (t \right )+9 y \left (t \right )+37 \end{align*}
✓ Maple. Time used: 0.108 (sec). Leaf size: 80
ode:=[diff(x(t),t) = -7*x(t)+10*y(t)+18*exp(t), diff(y(t),t) = -10*x(t)+9*y(t)+37];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= 10+\frac {{\mathrm e}^{t} \left (-20+4 \cos \left (6 t \right ) c_{1} -3 \cos \left (6 t \right ) c_{2} +3 \sin \left (6 t \right ) c_{1} +4 \sin \left (6 t \right ) c_{2} -20 \cos \left (6 t \right )-15 \sin \left (6 t \right )\right )}{5} \\
y &= 7+{\mathrm e}^{t} \left (-5+\cos \left (6 t \right ) c_{1} +\sin \left (6 t \right ) c_{2} -5 \cos \left (6 t \right )\right ) \\
\end{align*}
✓ Mathematica. Time used: 0.492 (sec). Leaf size: 260
ode={D[x[t],t]==-7*x[t]+10*y[t]+18*Exp[t],D[y[t],t]==-10*x[t]+9*y[t]+37};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to \frac {1}{3} e^t \left ((3 \cos (6 t)-4 \sin (6 t)) \int _1^t\left (18 \cos (6 K[1])+\frac {1}{3} \left (72-185 e^{-K[1]}\right ) \sin (6 K[1])\right )dK[1]+5 \sin (6 t) \int _1^t\left (\frac {37}{3} e^{-K[2]} (3 \cos (6 K[2])-4 \sin (6 K[2]))+30 \sin (6 K[2])\right )dK[2]+3 c_1 \cos (6 t)-4 c_1 \sin (6 t)+5 c_2 \sin (6 t)\right ) \\
y(t)\to \frac {1}{3} e^t \left (-5 \sin (6 t) \int _1^t\left (18 \cos (6 K[1])+\frac {1}{3} \left (72-185 e^{-K[1]}\right ) \sin (6 K[1])\right )dK[1]+(4 \sin (6 t)+3 \cos (6 t)) \int _1^t\left (\frac {37}{3} e^{-K[2]} (3 \cos (6 K[2])-4 \sin (6 K[2]))+30 \sin (6 K[2])\right )dK[2]+3 c_2 \cos (6 t)-5 c_1 \sin (6 t)+4 c_2 \sin (6 t)\right ) \\
\end{align*}
✓ Sympy. Time used: 0.366 (sec). Leaf size: 143
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(7*x(t) - 10*y(t) - 18*exp(t) + Derivative(x(t), t),0),Eq(10*x(t) - 9*y(t) + Derivative(y(t), t) - 37,0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = \left (\frac {3 C_{1}}{5} + \frac {4 C_{2}}{5}\right ) e^{t} \cos {\left (6 t \right )} - \left (\frac {4 C_{1}}{5} - \frac {3 C_{2}}{5}\right ) e^{t} \sin {\left (6 t \right )} - 4 e^{t} \sin ^{2}{\left (6 t \right )} - 4 e^{t} \cos ^{2}{\left (6 t \right )} + 10 \sin ^{2}{\left (6 t \right )} + 10 \cos ^{2}{\left (6 t \right )}, \ y{\left (t \right )} = - C_{1} e^{t} \sin {\left (6 t \right )} + C_{2} e^{t} \cos {\left (6 t \right )} - 5 e^{t} \sin ^{2}{\left (6 t \right )} - 5 e^{t} \cos ^{2}{\left (6 t \right )} + 7 \sin ^{2}{\left (6 t \right )} + 7 \cos ^{2}{\left (6 t \right )}\right ]
\]