85.88.17 problem 2 (h)
Internal
problem
ID
[23044]
Book
:
Applied
Differential
Equations.
By
Murray
R.
Spiegel.
3rd
edition.
1980.
Pearson.
ISBN
978-0130400970
Section
:
Chapter
10.
Systems
of
differential
equations
and
their
applications.
A
Exercises
at
page
499
Problem
number
:
2
(h)
Date
solved
:
Thursday, October 02, 2025 at 09:17:46 PM
CAS
classification
:
system_of_ODEs
\begin{align*} 2 \frac {d}{d t}x \left (t \right )-x \left (t \right )+7 \frac {d}{d t}y \left (t \right )+3 y \left (t \right )&=90 \sin \left (2 t \right )\\ \frac {d}{d t}x \left (t \right )-5 x \left (t \right )+8 \frac {d}{d t}y \left (t \right )-3 y \left (t \right )&=0 \end{align*}
✓ Maple. Time used: 0.323 (sec). Leaf size: 85
ode:=[2*diff(x(t),t)-x(t)+7*diff(y(t),t)+3*y(t) = 90*sin(2*t), diff(x(t),t)-5*x(t)+8*diff(y(t),t)-3*y(t) = 0];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= {\mathrm e}^{-t} \sin \left (t \right ) c_2 +{\mathrm e}^{-t} \cos \left (t \right ) c_1 +35 \sin \left (2 t \right )-10 \cos \left (2 t \right ) \\
y \left (t \right ) &= -\frac {2 \,{\mathrm e}^{-t} \sin \left (t \right ) c_2}{5}-\frac {{\mathrm e}^{-t} \cos \left (t \right ) c_2}{5}-\frac {2 \,{\mathrm e}^{-t} \cos \left (t \right ) c_1}{5}+\frac {{\mathrm e}^{-t} \sin \left (t \right ) c_1}{5}-8 \cos \left (2 t \right )-9 \sin \left (2 t \right ) \\
\end{align*}
✓ Mathematica. Time used: 0.005 (sec). Leaf size: 85
ode={2*D[x[t],t]-x[t]+7*D[y[t],t]+3*y[t]==90*Sin[2*t], D[x[t],t]-5*x[t]+8*D[y[t],t]-3*y[t]==0};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*} x(t)&\to 35 \sin (2 t)-10 \cos (2 t)-5 c_2 e^{-t} \sin (t)+c_1 e^{-t} (\cos (t)-2 \sin (t))\\ y(t)&\to -9 \sin (2 t)-8 \cos (2 t)+c_1 e^{-t} \sin (t)+c_2 e^{-t} (2 \sin (t)+\cos (t)) \end{align*}
✓ Sympy. Time used: 0.497 (sec). Leaf size: 139
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(-x(t) + 3*y(t) - 90*sin(2*t) + 2*Derivative(x(t), t) + 7*Derivative(y(t), t),0),Eq(-5*x(t) - 3*y(t) + Derivative(x(t), t) + 8*Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = - \left (C_{1} - 2 C_{2}\right ) e^{- t} \sin {\left (t \right )} - \left (2 C_{1} + C_{2}\right ) e^{- t} \cos {\left (t \right )} + 35 \sin ^{2}{\left (t \right )} \sin {\left (2 t \right )} - 10 \sin ^{2}{\left (t \right )} \cos {\left (2 t \right )} + 35 \sin {\left (2 t \right )} \cos ^{2}{\left (t \right )} - 10 \cos ^{2}{\left (t \right )} \cos {\left (2 t \right )}, \ y{\left (t \right )} = C_{1} e^{- t} \cos {\left (t \right )} - C_{2} e^{- t} \sin {\left (t \right )} - 9 \sin ^{2}{\left (t \right )} \sin {\left (2 t \right )} - 8 \sin ^{2}{\left (t \right )} \cos {\left (2 t \right )} - 9 \sin {\left (2 t \right )} \cos ^{2}{\left (t \right )} - 8 \cos ^{2}{\left (t \right )} \cos {\left (2 t \right )}\right ]
\]