67.7.10 problem Problem 5(b)
Internal
problem
ID
[14050]
Book
:
APPLIED
DIFFERENTIAL
EQUATIONS
The
Primary
Course
by
Vladimir
A.
Dobrushkin.
CRC
Press
2015
Section
:
Chapter
8.3
Systems
of
Linear
Differential
Equations
(Variation
of
Parameters).
Problems
page
514
Problem
number
:
Problem
5(b)
Date
solved
:
Monday, March 31, 2025 at 08:23:02 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=3 x \left (t \right )-2 y \left (t \right )+24 \sin \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=9 x \left (t \right )-3 y \left (t \right )+12 \cos \left (t \right ) \end{align*}
With initial conditions
\begin{align*} x \left (0\right ) = 1\\ y \left (0\right ) = -1 \end{align*}
✓ Maple. Time used: 0.273 (sec). Leaf size: 43
ode:=[diff(x(t),t) = 3*x(t)-2*y(t)+24*sin(t), diff(y(t),t) = 9*x(t)-3*y(t)+12*cos(t)];
ic:=x(0) = 1y(0) = -1;
dsolve([ode,ic]);
\begin{align*}
x \left (t \right ) &= -\frac {4 \sin \left (3 t \right )}{3}+\cos \left (3 t \right )+9 \sin \left (t \right ) \\
y \left (t \right ) &= \frac {7 \cos \left (3 t \right )}{2}-\frac {\sin \left (3 t \right )}{2}-\frac {9 \cos \left (t \right )}{2}+\frac {51 \sin \left (t \right )}{2} \\
\end{align*}
✓ Mathematica. Time used: 0.007 (sec). Leaf size: 50
ode={D[x[t],t]==3*x[t]-2*y[t]+24*Sin[t],D[y[t],t]==9*x[t]-3*y[t]+12*Cos[t]};
ic={x[0]==1,y[0]==-1};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to 9 \sin (t)-\frac {4}{3} \sin (3 t)+\cos (3 t) \\
y(t)\to \frac {1}{2} (51 \sin (t)-\sin (3 t)-9 \cos (t)+7 \cos (3 t)) \\
\end{align*}
✓ Sympy. Time used: 0.362 (sec). Leaf size: 119
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(-3*x(t) + 2*y(t) - 24*sin(t) + Derivative(x(t), t),0),Eq(-9*x(t) + 3*y(t) - 12*cos(t) + Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = \left (\frac {C_{1}}{3} - \frac {C_{2}}{3}\right ) \cos {\left (3 t \right )} - \left (\frac {C_{1}}{3} + \frac {C_{2}}{3}\right ) \sin {\left (3 t \right )} + 9 \sin {\left (t \right )} \sin ^{2}{\left (3 t \right )} + 9 \sin {\left (t \right )} \cos ^{2}{\left (3 t \right )}, \ y{\left (t \right )} = C_{1} \cos {\left (3 t \right )} - C_{2} \sin {\left (3 t \right )} + \frac {51 \sin {\left (t \right )} \sin ^{2}{\left (3 t \right )}}{2} + \frac {51 \sin {\left (t \right )} \cos ^{2}{\left (3 t \right )}}{2} - \frac {9 \sin ^{2}{\left (3 t \right )} \cos {\left (t \right )}}{2} - \frac {9 \cos {\left (t \right )} \cos ^{2}{\left (3 t \right )}}{2}\right ]
\]