76.19.22 problem 22
Internal
problem
ID
[17667]
Book
:
Differential
equations.
An
introduction
to
modern
methods
and
applications.
James
Brannan,
William
E.
Boyce.
Third
edition.
Wiley
2015
Section
:
Chapter
5.
The
Laplace
transform.
Section
5.4
(Solving
differential
equations
with
Laplace
transform).
Problems
at
page
327
Problem
number
:
22
Date
solved
:
Thursday, March 13, 2025 at 10:46:13 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}y_{1} \left (t \right )&=-y_{1} \left (t \right )-5 y_{2} \left (t \right )+3\\ \frac {d}{d t}y_{2} \left (t \right )&=y_{1} \left (t \right )+3 y_{2} \left (t \right )+5 \cos \left (t \right ) \end{align*}
With initial conditions
\begin{align*} y_{1} \left (0\right ) = 1\\ y_{2} \left (0\right ) = -1 \end{align*}
✓ Maple. Time used: 0.325 (sec). Leaf size: 51
ode:=[diff(y__1(t),t) = -y__1(t)-5*y__2(t)+3, diff(y__2(t),t) = y__1(t)+3*y__2(t)+5*cos(t)];
ic:=y__1(0) = 1y__2(0) = -1;
dsolve([ode,ic]);
\begin{align*}
y_{1} \left (t \right ) &= -\frac {27 \,{\mathrm e}^{t} \sin \left (t \right )}{2}+\frac {21 \,{\mathrm e}^{t} \cos \left (t \right )}{2}+10 \sin \left (t \right )-5 \cos \left (t \right )-\frac {9}{2} \\
y_{2} \left (t \right ) &= \frac {15 \,{\mathrm e}^{t} \sin \left (t \right )}{2}-\frac {3 \,{\mathrm e}^{t} \cos \left (t \right )}{2}-\cos \left (t \right )-3 \sin \left (t \right )+\frac {3}{2} \\
\end{align*}
✓ Mathematica. Time used: 0.113 (sec). Leaf size: 62
ode={D[y1[t],t]==-1*y1[t]-5*y2[t]+3,D[y2[t],t]==1*y1[t]+3*y2[t]+5*Cos[t]};
ic={y1[0]==1,y2[0]==-1};
DSolve[{ode,ic},{y1[t],y2[t]},t,IncludeSingularSolutions->True]
\begin{align*}
\text {y1}(t)\to \frac {1}{2} \left (\left (20-27 e^t\right ) \sin (t)+\left (21 e^t-10\right ) \cos (t)-9\right ) \\
\text {y2}(t)\to \frac {1}{2} \left (3 \left (5 e^t-2\right ) \sin (t)-\left (\left (3 e^t+2\right ) \cos (t)\right )+3\right ) \\
\end{align*}
✓ Sympy. Time used: 0.413 (sec). Leaf size: 143
from sympy import *
t = symbols("t")
y__1 = Function("y__1")
y__2 = Function("y__2")
ode=[Eq(y__1(t) + 5*y__2(t) + Derivative(y__1(t), t) - 3,0),Eq(-y__1(t) - 3*y__2(t) - 5*cos(t) + Derivative(y__2(t), t),0)]
ics = {}
dsolve(ode,func=[y__1(t),y__2(t)],ics=ics)
\[
\left [ y^{1}{\left (t \right )} = - \left (C_{1} - 2 C_{2}\right ) e^{t} \sin {\left (t \right )} - \left (2 C_{1} + C_{2}\right ) e^{t} \cos {\left (t \right )} + 10 \sin ^{3}{\left (t \right )} - 5 \sin ^{2}{\left (t \right )} \cos {\left (t \right )} - \frac {9 \sin ^{2}{\left (t \right )}}{2} + 10 \sin {\left (t \right )} \cos ^{2}{\left (t \right )} - 5 \cos ^{3}{\left (t \right )} - \frac {9 \cos ^{2}{\left (t \right )}}{2}, \ y^{2}{\left (t \right )} = C_{1} e^{t} \cos {\left (t \right )} - C_{2} e^{t} \sin {\left (t \right )} - 3 \sin ^{3}{\left (t \right )} - \sin ^{2}{\left (t \right )} \cos {\left (t \right )} + \frac {3 \sin ^{2}{\left (t \right )}}{2} - 3 \sin {\left (t \right )} \cos ^{2}{\left (t \right )} - \cos ^{3}{\left (t \right )} + \frac {3 \cos ^{2}{\left (t \right )}}{2}\right ]
\]