76.6.2 problem 2
Internal
problem
ID
[17386]
Book
:
Differential
equations.
An
introduction
to
modern
methods
and
applications.
James
Brannan,
William
E.
Boyce.
Third
edition.
Wiley
2015
Section
:
Chapter
3.
Systems
of
two
first
order
equations.
Section
3.2
(Two
first
order
linear
differential
equations).
Problems
at
page
142
Problem
number
:
2
Date
solved
:
Monday, March 31, 2025 at 04:12:33 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=x \left (t \right )+2 y \left (t \right )+\sin \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-x \left (t \right )+y \left (t \right )-\cos \left (t \right ) \end{align*}
✓ Maple. Time used: 0.881 (sec). Leaf size: 69
ode:=[diff(x(t),t) = x(t)+2*y(t)+sin(t), diff(y(t),t) = -x(t)+y(t)-cos(t)];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= {\mathrm e}^{t} \sin \left (\sqrt {2}\, t \right ) c_2 +{\mathrm e}^{t} \cos \left (\sqrt {2}\, t \right ) c_1 -\frac {\cos \left (t \right )}{2} \\
y \left (t \right ) &= \frac {{\mathrm e}^{t} \sqrt {2}\, \cos \left (\sqrt {2}\, t \right ) c_2}{2}-\frac {{\mathrm e}^{t} \sqrt {2}\, \sin \left (\sqrt {2}\, t \right ) c_1}{2}-\frac {\sin \left (t \right )}{4}+\frac {\cos \left (t \right )}{4} \\
\end{align*}
✓ Mathematica. Time used: 0.229 (sec). Leaf size: 317
ode={D[x[t],t]==x[t]+2*y[t]+Sin[t],D[y[t],t]==-x[t]+y[t]-Cos[t]};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to e^t \left (\cos \left (\sqrt {2} t\right ) \int _1^te^{-K[1]} \left (\cos \left (\sqrt {2} K[1]\right ) \sin (K[1])+\sqrt {2} \cos (K[1]) \sin \left (\sqrt {2} K[1]\right )\right )dK[1]+\sqrt {2} \sin \left (\sqrt {2} t\right ) \int _1^t\frac {1}{2} e^{-K[2]} \left (\sqrt {2} \sin (K[2]) \sin \left (\sqrt {2} K[2]\right )-2 \cos (K[2]) \cos \left (\sqrt {2} K[2]\right )\right )dK[2]+c_1 \cos \left (\sqrt {2} t\right )+\sqrt {2} c_2 \sin \left (\sqrt {2} t\right )\right ) \\
y(t)\to \frac {1}{2} e^t \left (2 \cos \left (\sqrt {2} t\right ) \int _1^t\frac {1}{2} e^{-K[2]} \left (\sqrt {2} \sin (K[2]) \sin \left (\sqrt {2} K[2]\right )-2 \cos (K[2]) \cos \left (\sqrt {2} K[2]\right )\right )dK[2]-\sqrt {2} \sin \left (\sqrt {2} t\right ) \int _1^te^{-K[1]} \left (\cos \left (\sqrt {2} K[1]\right ) \sin (K[1])+\sqrt {2} \cos (K[1]) \sin \left (\sqrt {2} K[1]\right )\right )dK[1]+2 c_2 \cos \left (\sqrt {2} t\right )-\sqrt {2} c_1 \sin \left (\sqrt {2} t\right )\right ) \\
\end{align*}
✓ Sympy. Time used: 1.222 (sec). Leaf size: 160
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(-x(t) - 2*y(t) - sin(t) + Derivative(x(t), t),0),Eq(x(t) - y(t) + cos(t) + Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = \sqrt {2} C_{1} e^{t} \sin {\left (\sqrt {2} t \right )} + \sqrt {2} C_{2} e^{t} \cos {\left (\sqrt {2} t \right )} - \frac {\sin ^{2}{\left (\sqrt {2} t \right )} \cos {\left (t \right )}}{2} - \frac {\cos {\left (t \right )} \cos ^{2}{\left (\sqrt {2} t \right )}}{2}, \ y{\left (t \right )} = C_{1} e^{t} \cos {\left (\sqrt {2} t \right )} - C_{2} e^{t} \sin {\left (\sqrt {2} t \right )} - \frac {\sin {\left (t \right )} \sin ^{2}{\left (\sqrt {2} t \right )}}{4} - \frac {\sin {\left (t \right )} \cos ^{2}{\left (\sqrt {2} t \right )}}{4} + \frac {\sin ^{2}{\left (\sqrt {2} t \right )} \cos {\left (t \right )}}{4} + \frac {\cos {\left (t \right )} \cos ^{2}{\left (\sqrt {2} t \right )}}{4}\right ]
\]