65.14.5 problem 26.1 (v)
Internal
problem
ID
[13742]
Book
:
AN
INTRODUCTION
TO
ORDINARY
DIFFERENTIAL
EQUATIONS
by
JAMES
C.
ROBINSON.
Cambridge
University
Press
2004
Section
:
Chapter
26,
Explicit
solutions
of
coupled
linear
systems.
Exercises
page
257
Problem
number
:
26.1
(v)
Date
solved
:
Wednesday, March 05, 2025 at 10:14:47 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=2 x \left (t \right )+5 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-2 x \left (t \right )+\cos \left (3 t \right ) \end{align*}
With initial conditions
\begin{align*} x \left (0\right ) = 2\\ y \left (0\right ) = -1 \end{align*}
✓ Maple. Time used: 0.276 (sec). Leaf size: 65
ode:=[diff(x(t),t) = 2*x(t)+5*y(t), diff(y(t),t) = -2*x(t)+cos(3*t)];
ic:=x(0) = 2y(0) = -1;
dsolve([ode,ic]);
\begin{align*}
x \left (t \right ) &= -\frac {16 \,{\mathrm e}^{t} \sin \left (3 t \right )}{111}+\frac {69 \,{\mathrm e}^{t} \cos \left (3 t \right )}{37}-\frac {30 \sin \left (3 t \right )}{37}+\frac {5 \cos \left (3 t \right )}{37} \\
y \left (t \right ) &= -\frac {121 \,{\mathrm e}^{t} \sin \left (3 t \right )}{111}-\frac {17 \,{\mathrm e}^{t} \cos \left (3 t \right )}{37}-\frac {20 \cos \left (3 t \right )}{37}+\frac {9 \sin \left (3 t \right )}{37} \\
\end{align*}
✓ Mathematica. Time used: 0.209 (sec). Leaf size: 418
ode={D[x[t],t]==2*x[t]+5*y[t],D[y[t],t]==-2*x[t]+Cos[3*t]};
ic={x[0]==2,y[0]==-1};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to \frac {1}{3} e^t \left (-(\sin (3 t)+3 \cos (3 t)) \int _1^0-\frac {5}{6} e^{-K[1]} \sin (6 K[1])dK[1]+(\sin (3 t)+3 \cos (3 t)) \int _1^t-\frac {5}{6} e^{-K[1]} \sin (6 K[1])dK[1]-5 \sin (3 t) \int _1^0\frac {1}{3} e^{-K[2]} \cos (3 K[2]) (3 \cos (3 K[2])+\sin (3 K[2]))dK[2]+5 \sin (3 t) \int _1^t\frac {1}{3} e^{-K[2]} \cos (3 K[2]) (3 \cos (3 K[2])+\sin (3 K[2]))dK[2]-3 \sin (3 t)+6 \cos (3 t)\right ) \\
y(t)\to -\frac {1}{3} e^t \left (-2 \sin (3 t) \int _1^0-\frac {5}{6} e^{-K[1]} \sin (6 K[1])dK[1]+2 \sin (3 t) \int _1^t-\frac {5}{6} e^{-K[1]} \sin (6 K[1])dK[1]+3 \cos (3 t) \int _1^0\frac {1}{3} e^{-K[2]} \cos (3 K[2]) (3 \cos (3 K[2])+\sin (3 K[2]))dK[2]-3 \cos (3 t) \int _1^t\frac {1}{3} e^{-K[2]} \cos (3 K[2]) (3 \cos (3 K[2])+\sin (3 K[2]))dK[2]-\sin (3 t) \int _1^0\frac {1}{3} e^{-K[2]} \cos (3 K[2]) (3 \cos (3 K[2])+\sin (3 K[2]))dK[2]+\sin (3 t) \int _1^t\frac {1}{3} e^{-K[2]} \cos (3 K[2]) (3 \cos (3 K[2])+\sin (3 K[2]))dK[2]+3 \sin (3 t)+3 \cos (3 t)\right ) \\
\end{align*}
✓ Sympy. Time used: 0.428 (sec). Leaf size: 160
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(-2*x(t) - 5*y(t) + Derivative(x(t), t),0),Eq(2*x(t) - cos(3*t) + Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = \left (\frac {C_{1}}{2} + \frac {3 C_{2}}{2}\right ) e^{t} \sin {\left (3 t \right )} + \left (\frac {3 C_{1}}{2} - \frac {C_{2}}{2}\right ) e^{t} \cos {\left (3 t \right )} - \frac {30 \sin ^{3}{\left (3 t \right )}}{37} + \frac {5 \sin ^{2}{\left (3 t \right )} \cos {\left (3 t \right )}}{37} - \frac {30 \sin {\left (3 t \right )} \cos ^{2}{\left (3 t \right )}}{37} + \frac {5 \cos ^{3}{\left (3 t \right )}}{37}, \ y{\left (t \right )} = - C_{1} e^{t} \sin {\left (3 t \right )} + C_{2} e^{t} \cos {\left (3 t \right )} + \frac {9 \sin ^{3}{\left (3 t \right )}}{37} - \frac {20 \sin ^{2}{\left (3 t \right )} \cos {\left (3 t \right )}}{37} + \frac {9 \sin {\left (3 t \right )} \cos ^{2}{\left (3 t \right )}}{37} - \frac {20 \cos ^{3}{\left (3 t \right )}}{37}\right ]
\]