76.6.2 problem 2
Internal
problem
ID
[17457]
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
:
Tuesday, January 28, 2025 at 10:39:00 AM
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*}
✓ Solution by Maple
Time used: 3.128 (sec). Leaf size: 69
dsolve([diff(x(t),t)=x(t)+2*y(t)+sin(t),diff(y(t),t)=-x(t)+y(t)-cos(t)],singsol=all)
\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*}
✓ Solution by Mathematica
Time used: 0.229 (sec). Leaf size: 317
DSolve[{D[x[t],t]==x[t]+2*y[t]+Sin[t],D[y[t],t]==-x[t]+y[t]-Cos[t]},{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*}