65.14.2 problem 26.1 (ii)
Internal
problem
ID
[13739]
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
(ii)
Date
solved
:
Wednesday, March 05, 2025 at 10:14:43 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=x \left (t \right )-4 y \left (t \right )+\cos \left (2 t \right )\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right )+y \left (t \right ) \end{align*}
With initial conditions
\begin{align*} x \left (0\right ) = 1\\ y \left (0\right ) = 1 \end{align*}
✓ Maple. Time used: 0.299 (sec). Leaf size: 65
ode:=[diff(x(t),t) = x(t)-4*y(t)+cos(2*t), diff(y(t),t) = x(t)+y(t)];
ic:=x(0) = 1y(0) = 1;
dsolve([ode,ic]);
\begin{align*}
x \left (t \right ) &= \frac {26 \cos \left (2 t \right ) {\mathrm e}^{t}}{17}-\frac {32 \sin \left (2 t \right ) {\mathrm e}^{t}}{17}+\frac {2 \sin \left (2 t \right )}{17}-\frac {9 \cos \left (2 t \right )}{17} \\
y \left (t \right ) &= \frac {13 \sin \left (2 t \right ) {\mathrm e}^{t}}{17}+\frac {16 \cos \left (2 t \right ) {\mathrm e}^{t}}{17}+\frac {\cos \left (2 t \right )}{17}-\frac {4 \sin \left (2 t \right )}{17} \\
\end{align*}
✓ Mathematica. Time used: 0.085 (sec). Leaf size: 259
ode={D[x[t],t]==x[t]-4*y[t]+Cos[2*t],D[y[t],t]==x[t]+y[t]};
ic={x[0]==1,y[0]==1};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to e^t \left (2 \sin (2 t) \int _1^0-\frac {1}{4} e^{-K[2]} \sin (4 K[2])dK[2]-2 \sin (2 t) \int _1^t-\frac {1}{4} e^{-K[2]} \sin (4 K[2])dK[2]+\cos (2 t) \left (-\int _1^0e^{-K[1]} \cos ^2(2 K[1])dK[1]\right )+\cos (2 t) \int _1^te^{-K[1]} \cos ^2(2 K[1])dK[1]-2 \sin (2 t)+\cos (2 t)\right ) \\
y(t)\to \frac {1}{2} e^t \left (-\sin (2 t) \int _1^0e^{-K[1]} \cos ^2(2 K[1])dK[1]+\sin (2 t) \int _1^te^{-K[1]} \cos ^2(2 K[1])dK[1]-2 \cos (2 t) \int _1^0-\frac {1}{4} e^{-K[2]} \sin (4 K[2])dK[2]+2 \cos (2 t) \int _1^t-\frac {1}{4} e^{-K[2]} \sin (4 K[2])dK[2]+\sin (2 t)+2 \cos (2 t)\right ) \\
\end{align*}
✓ Sympy. Time used: 0.353 (sec). Leaf size: 146
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(-x(t) + 4*y(t) - cos(2*t) + Derivative(x(t), t),0),Eq(-x(t) - y(t) + Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = - 2 C_{1} e^{t} \sin {\left (2 t \right )} - 2 C_{2} e^{t} \cos {\left (2 t \right )} + \frac {2 \sin ^{3}{\left (2 t \right )}}{17} - \frac {9 \sin ^{2}{\left (2 t \right )} \cos {\left (2 t \right )}}{17} + \frac {2 \sin {\left (2 t \right )} \cos ^{2}{\left (2 t \right )}}{17} - \frac {9 \cos ^{3}{\left (2 t \right )}}{17}, \ y{\left (t \right )} = C_{1} e^{t} \cos {\left (2 t \right )} - C_{2} e^{t} \sin {\left (2 t \right )} - \frac {4 \sin ^{3}{\left (2 t \right )}}{17} + \frac {\sin ^{2}{\left (2 t \right )} \cos {\left (2 t \right )}}{17} - \frac {4 \sin {\left (2 t \right )} \cos ^{2}{\left (2 t \right )}}{17} + \frac {\cos ^{3}{\left (2 t \right )}}{17}\right ]
\]