63.19.7 problem 3(a)
Internal
problem
ID
[13142]
Book
:
A
First
Course
in
Differential
Equations
by
J.
David
Logan.
Third
Edition.
Springer-Verlag,
NY.
2015.
Section
:
Chapter
4,
Linear
Systems.
Exercises
page
202
Problem
number
:
3(a)
Date
solved
:
Wednesday, March 05, 2025 at 09:18:09 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=2 x \left (t \right )+3 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-x \left (t \right )-14 \end{align*}
✓ Maple. Time used: 0.065 (sec). Leaf size: 76
ode:=[diff(x(t),t) = 2*x(t)+3*y(t), diff(y(t),t) = -x(t)-14];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= -14+{\mathrm e}^{t} \left (\sqrt {2}\, \sin \left (\sqrt {2}\, t \right ) c_{1} -\sqrt {2}\, \cos \left (\sqrt {2}\, t \right ) c_{2} -\sin \left (\sqrt {2}\, t \right ) c_{2} -\cos \left (\sqrt {2}\, t \right ) c_{1} \right ) \\
y &= \frac {28}{3}+{\mathrm e}^{t} \left (\sin \left (\sqrt {2}\, t \right ) c_{2} +\cos \left (\sqrt {2}\, t \right ) c_{1} \right ) \\
\end{align*}
✓ Mathematica. Time used: 0.167 (sec). Leaf size: 336
ode={D[x[t],t]==2*x[t]+3*y[t],D[y[t],t]==-x[t]-14};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to \frac {1}{2} e^t \left (\left (\sqrt {2} \sin \left (\sqrt {2} t\right )+2 \cos \left (\sqrt {2} t\right )\right ) \int _1^t21 \sqrt {2} e^{-K[1]} \sin \left (\sqrt {2} K[1]\right )dK[1]+3 \sqrt {2} \sin \left (\sqrt {2} t\right ) \int _1^t-7 e^{-K[2]} \left (2 \cos \left (\sqrt {2} K[2]\right )+\sqrt {2} \sin \left (\sqrt {2} K[2]\right )\right )dK[2]+2 c_1 \cos \left (\sqrt {2} t\right )+\sqrt {2} c_1 \sin \left (\sqrt {2} t\right )+3 \sqrt {2} c_2 \sin \left (\sqrt {2} t\right )\right ) \\
y(t)\to -\frac {1}{2} e^t \left (\sqrt {2} \sin \left (\sqrt {2} t\right ) \int _1^t21 \sqrt {2} e^{-K[1]} \sin \left (\sqrt {2} K[1]\right )dK[1]+\left (\sqrt {2} \sin \left (\sqrt {2} t\right )-2 \cos \left (\sqrt {2} t\right )\right ) \int _1^t-7 e^{-K[2]} \left (2 \cos \left (\sqrt {2} K[2]\right )+\sqrt {2} \sin \left (\sqrt {2} K[2]\right )\right )dK[2]-2 c_2 \cos \left (\sqrt {2} t\right )+\sqrt {2} c_1 \sin \left (\sqrt {2} t\right )+\sqrt {2} c_2 \sin \left (\sqrt {2} t\right )\right ) \\
\end{align*}
✓ Sympy. Time used: 0.368 (sec). Leaf size: 122
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(-2*x(t) - 3*y(t) + Derivative(x(t), t),0),Eq(x(t) + Derivative(y(t), t) + 14,0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = \left (C_{1} + \sqrt {2} C_{2}\right ) e^{t} \sin {\left (\sqrt {2} t \right )} + \left (\sqrt {2} C_{1} - C_{2}\right ) e^{t} \cos {\left (\sqrt {2} t \right )} - 14 \sin ^{2}{\left (\sqrt {2} t \right )} - 14 \cos ^{2}{\left (\sqrt {2} t \right )}, \ y{\left (t \right )} = - C_{1} e^{t} \sin {\left (\sqrt {2} t \right )} + C_{2} e^{t} \cos {\left (\sqrt {2} t \right )} + \frac {28 \sin ^{2}{\left (\sqrt {2} t \right )}}{3} + \frac {28 \cos ^{2}{\left (\sqrt {2} t \right )}}{3}\right ]
\]