85.92.11 problem 2 (e)
Internal
problem
ID
[23060]
Book
:
Applied
Differential
Equations.
By
Murray
R.
Spiegel.
3rd
edition.
1980.
Pearson.
ISBN
978-0130400970
Section
:
Chapter
11.
Matrix
eigenvalue
methods
for
systems
of
linear
differential
equations.
A
Exercises
at
page
528
Problem
number
:
2
(e)
Date
solved
:
Thursday, October 02, 2025 at 09:18:27 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )-3 x \left (t \right )-6 y \left (t \right )&=9-9 t\\ \frac {d}{d t}y \left (t \right )+3 x \left (t \right )+3 y \left (t \right )&=9 t \,{\mathrm e}^{-3 t} \end{align*}
✓ Maple. Time used: 0.257 (sec). Leaf size: 81
ode:=[diff(x(t),t)-3*x(t)-6*y(t) = 9-9*t, diff(y(t),t)+3*x(t)+3*y(t) = 9*exp(-3*t)*t];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= \cos \left (3 t \right ) c_1 +\sin \left (3 t \right ) c_2 +3 t \,{\mathrm e}^{-3 t}+{\mathrm e}^{-3 t}-3 t +2 \\
y \left (t \right ) &= -\frac {\cos \left (3 t \right ) c_1}{2}+\frac {\cos \left (3 t \right ) c_2}{2}-\frac {\sin \left (3 t \right ) c_1}{2}-\frac {\sin \left (3 t \right ) c_2}{2}-3 t \,{\mathrm e}^{-3 t}-\frac {{\mathrm e}^{-3 t}}{2}+3 t -3 \\
\end{align*}
✓ Mathematica. Time used: 0.624 (sec). Leaf size: 85
ode={D[x[t],{t,1}]-3*x[t]-6*y[t]==9*(1-t), D[y[t],{t,1}]+3*x[t]+3*y[t]==9*t*Exp[-3*t]};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*} x(t)&\to 3 e^{-3 t} t-3 t+e^{-3 t}+c_1 \cos (3 t)+(c_1+2 c_2) \sin (3 t)+2\\ y(t)&\to -3 e^{-3 t} t+3 t-\frac {e^{-3 t}}{2}+c_2 \cos (3 t)-(c_1+c_2) \sin (3 t)-3 \end{align*}
✓ Sympy. Time used: 0.346 (sec). Leaf size: 221
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(9*t - 3*x(t) - 6*y(t) + Derivative(x(t), t) - 9,0),Eq(-9*t*exp(-3*t) + 3*x(t) + 3*y(t) + Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = - 3 t \sin ^{2}{\left (3 t \right )} - 3 t \cos ^{2}{\left (3 t \right )} + 3 t e^{- 3 t} \sin ^{2}{\left (3 t \right )} + 3 t e^{- 3 t} \cos ^{2}{\left (3 t \right )} - \left (C_{1} - C_{2}\right ) \cos {\left (3 t \right )} + \left (C_{1} + C_{2}\right ) \sin {\left (3 t \right )} + 2 \sin ^{2}{\left (3 t \right )} + 2 \cos ^{2}{\left (3 t \right )} + e^{- 3 t} \sin ^{2}{\left (3 t \right )} + e^{- 3 t} \cos ^{2}{\left (3 t \right )}, \ y{\left (t \right )} = C_{1} \cos {\left (3 t \right )} - C_{2} \sin {\left (3 t \right )} + 3 t \sin ^{2}{\left (3 t \right )} + 3 t \cos ^{2}{\left (3 t \right )} - 3 t e^{- 3 t} \sin ^{2}{\left (3 t \right )} - 3 t e^{- 3 t} \cos ^{2}{\left (3 t \right )} - 3 \sin ^{2}{\left (3 t \right )} - 3 \cos ^{2}{\left (3 t \right )} - \frac {e^{- 3 t} \sin ^{2}{\left (3 t \right )}}{2} - \frac {e^{- 3 t} \cos ^{2}{\left (3 t \right )}}{2}\right ]
\]