67.7.1 problem Problem 3(a)
Internal
problem
ID
[14033]
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
3(a)
Date
solved
:
Wednesday, March 05, 2025 at 10:26:34 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=-4 x \left (t \right )+9 y \left (t \right )+12 \,{\mathrm e}^{-t}\\ \frac {d}{d t}y \left (t \right )&=-5 x \left (t \right )+2 y \left (t \right ) \end{align*}
✓ Maple. Time used: 0.058 (sec). Leaf size: 65
ode:=[diff(x(t),t) = -4*x(t)+9*y(t)+12*exp(-t), diff(y(t),t) = -5*x(t)+2*y(t)];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= \frac {{\mathrm e}^{-t} \left (3 \cos \left (6 t \right ) c_{1} -6 \cos \left (6 t \right ) c_{2} +6 \sin \left (6 t \right ) c_{1} +3 \sin \left (6 t \right ) c_{2} -5\right )}{5} \\
y &= \frac {{\mathrm e}^{-t} \left (-5+3 \cos \left (6 t \right ) c_{1} +3 \sin \left (6 t \right ) c_{2} \right )}{3} \\
\end{align*}
✓ Mathematica. Time used: 0.046 (sec). Leaf size: 187
ode={D[x[t],t]==-4*x[t]+9*y[t]+12*Exp[-t],D[y[t],t]==-5*x[t]+2*y[t]};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to \frac {1}{2} e^{-t} \left (3 \sin (6 t) \int _1^t10 \sin (6 K[2])dK[2]+(2 \cos (6 t)-\sin (6 t)) \int _1^t6 (2 \cos (6 K[1])+\sin (6 K[1]))dK[1]+2 c_1 \cos (6 t)-c_1 \sin (6 t)+3 c_2 \sin (6 t)\right ) \\
y(t)\to \frac {1}{6} e^{-t} \left (-5 \sin (6 t) \int _1^t6 (2 \cos (6 K[1])+\sin (6 K[1]))dK[1]+3 (\sin (6 t)+2 \cos (6 t)) \int _1^t10 \sin (6 K[2])dK[2]+6 c_2 \cos (6 t)-5 c_1 \sin (6 t)+3 c_2 \sin (6 t)\right ) \\
\end{align*}
✓ Sympy. Time used: 0.182 (sec). Leaf size: 109
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(4*x(t) - 9*y(t) + Derivative(x(t), t) - 12*exp(-t),0),Eq(5*x(t) - 2*y(t) + Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = - \left (\frac {3 C_{1}}{5} - \frac {6 C_{2}}{5}\right ) e^{- t} \sin {\left (6 t \right )} + \left (\frac {6 C_{1}}{5} + \frac {3 C_{2}}{5}\right ) e^{- t} \cos {\left (6 t \right )} - e^{- t} \sin ^{2}{\left (6 t \right )} - e^{- t} \cos ^{2}{\left (6 t \right )}, \ y{\left (t \right )} = - C_{1} e^{- t} \sin {\left (6 t \right )} + C_{2} e^{- t} \cos {\left (6 t \right )} - \frac {5 e^{- t} \sin ^{2}{\left (6 t \right )}}{3} - \frac {5 e^{- t} \cos ^{2}{\left (6 t \right )}}{3}\right ]
\]