63.23.3 problem 5
Internal
problem
ID
[13164]
Book
:
A
First
Course
in
Differential
Equations
by
J.
David
Logan.
Third
Edition.
Springer-Verlag,
NY.
2015.
Section
:
Chapter
4,
Linear
Systems.
Exercises
page
244
Problem
number
:
5
Date
solved
:
Wednesday, March 05, 2025 at 09:18:33 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-x \left (t \right )+\cos \left (w t \right ) \end{align*}
✓ Maple. Time used: 1.036 (sec). Leaf size: 71
ode:=[diff(x(t),t) = y(t), diff(y(t),t) = -x(t)+cos(w*t)];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= c_{2} \sin \left (t \right )+\cos \left (t \right ) c_{1} -\frac {\cos \left (t w \right )}{w^{2}-1} \\
y &= \frac {\cos \left (t \right ) c_{2} w^{2}-\sin \left (t \right ) c_{1} w^{2}+w \sin \left (t w \right )-c_{2} \cos \left (t \right )+c_{1} \sin \left (t \right )}{\left (-1+w \right ) \left (1+w \right )} \\
\end{align*}
✓ Mathematica. Time used: 0.093 (sec). Leaf size: 110
ode={D[x[t],t]==0*x[t]+y[t],D[y[t],t]==-x[t]+Cos[w*t]};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to \cos (t) \int _1^t-\cos (w K[1]) \sin (K[1])dK[1]+\sin (t) \int _1^t\cos (K[2]) \cos (w K[2])dK[2]+c_1 \cos (t)+c_2 \sin (t) \\
y(t)\to \cos (t) \int _1^t\cos (K[2]) \cos (w K[2])dK[2]-\sin (t) \int _1^t-\cos (w K[1]) \sin (K[1])dK[1]+c_2 \cos (t)-c_1 \sin (t) \\
\end{align*}
✓ Sympy. Time used: 0.405 (sec). Leaf size: 235
from sympy import *
t = symbols("t")
w = symbols("w")
x = Function("x")
y = Function("y")
ode=[Eq(-y(t) + Derivative(x(t), t),0),Eq(x(t) - cos(t*w) + Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = C_{1} \sin {\left (t \right )} + C_{2} \cos {\left (t \right )} + \left (\begin {cases} \frac {t \sin ^{2}{\left (t \right )}}{2} + \frac {t \cos ^{2}{\left (t \right )}}{2} + \frac {\sin {\left (t \right )} \cos {\left (t \right )}}{2} & \text {for}\: w = -1 \vee w = 1 \\\frac {w \sin {\left (t w \right )} \cos {\left (t \right )}}{w^{2} - 1} - \frac {\sin {\left (t \right )} \cos {\left (t w \right )}}{w^{2} - 1} & \text {otherwise} \end {cases}\right ) \sin {\left (t \right )} - \left (\begin {cases} - \frac {\cos ^{2}{\left (t \right )}}{2} & \text {for}\: w = -1 \vee w = 1 \\\frac {w \sin {\left (t \right )} \sin {\left (t w \right )}}{w^{2} - 1} + \frac {\cos {\left (t \right )} \cos {\left (t w \right )}}{w^{2} - 1} & \text {otherwise} \end {cases}\right ) \cos {\left (t \right )}, \ y{\left (t \right )} = C_{1} \cos {\left (t \right )} - C_{2} \sin {\left (t \right )} + \left (\begin {cases} \frac {t \sin ^{2}{\left (t \right )}}{2} + \frac {t \cos ^{2}{\left (t \right )}}{2} + \frac {\sin {\left (t \right )} \cos {\left (t \right )}}{2} & \text {for}\: w = -1 \vee w = 1 \\\frac {w \sin {\left (t w \right )} \cos {\left (t \right )}}{w^{2} - 1} - \frac {\sin {\left (t \right )} \cos {\left (t w \right )}}{w^{2} - 1} & \text {otherwise} \end {cases}\right ) \cos {\left (t \right )} + \left (\begin {cases} - \frac {\cos ^{2}{\left (t \right )}}{2} & \text {for}\: w = -1 \vee w = 1 \\\frac {w \sin {\left (t \right )} \sin {\left (t w \right )}}{w^{2} - 1} + \frac {\cos {\left (t \right )} \cos {\left (t w \right )}}{w^{2} - 1} & \text {otherwise} \end {cases}\right ) \sin {\left (t \right )}\right ]
\]