82.56.2 problem Ex. 2
Internal
problem
ID
[18963]
Book
:
Introductory
Course
On
Differential
Equations
by
Daniel
A
Murray.
Longmans
Green
and
Co.
NY.
1924
Section
:
Chapter
XI.
Ordinary
differential
equations
with
more
than
two
variables.
End
of
chapter
problems
at
page
143
Problem
number
:
Ex.
2
Date
solved
:
Thursday, March 13, 2025 at 01:13:29 PM
CAS
classification
:
system_of_ODEs
\begin{align*} 4 \frac {d}{d t}x \left (t \right )+9 \frac {d}{d t}y \left (t \right )+2 x \left (t \right )+31 y \left (t \right )&={\mathrm e}^{t}\\ 3 \frac {d}{d t}x \left (t \right )+7 \frac {d}{d t}y \left (t \right )+x \left (t \right )+24 y \left (t \right )&=3 \end{align*}
✓ Maple. Time used: 0.079 (sec). Leaf size: 70
ode:=[4*diff(x(t),t)+9*diff(y(t),t)+2*x(t)+31*y(t) = exp(t), 3*diff(x(t),t)+7*diff(y(t),t)+x(t)+24*y(t) = 3];
dsolve(ode);
\begin{align*}
x &= {\mathrm e}^{-4 t} \sin \left (t \right ) c_{2} +{\mathrm e}^{-4 t} \cos \left (t \right ) c_{1} -\frac {93}{17}+\frac {31 \,{\mathrm e}^{t}}{26} \\
y &= -{\mathrm e}^{-4 t} \sin \left (t \right ) c_{2} -{\mathrm e}^{-4 t} \cos \left (t \right ) c_{2} -{\mathrm e}^{-4 t} \cos \left (t \right ) c_{1} +{\mathrm e}^{-4 t} \sin \left (t \right ) c_{1} -\frac {2 \,{\mathrm e}^{t}}{13}+\frac {6}{17} \\
\end{align*}
✓ Mathematica. Time used: 0.215 (sec). Leaf size: 79
ode={4*D[x[t],t]+9*D[y[t],t]+2*x[t]+31*y[t]==Exp[t],3*D[x[t],t]+7*D[y[t],t]+x[t]+24*y[t]==3};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to \frac {31 e^t}{26}+c_1 e^{-4 t} \cos (t)-(c_1+c_2) e^{-4 t} \sin (t)-\frac {93}{17} \\
y(t)\to -\frac {2 e^t}{13}+c_2 e^{-4 t} \cos (t)+(2 c_1+c_2) e^{-4 t} \sin (t)+\frac {6}{17} \\
\end{align*}
✓ Sympy. Time used: 3.786 (sec). Leaf size: 136
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(2*x(t) + 31*y(t) - exp(t) + 4*Derivative(x(t), t) + 9*Derivative(y(t), t),0),Eq(x(t) + 24*y(t) + 3*Derivative(x(t), t) + 7*Derivative(y(t), t) - 3,0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = - \left (\frac {C_{1}}{2} - \frac {C_{2}}{2}\right ) e^{- 4 t} \sin {\left (t \right )} - \left (\frac {C_{1}}{2} + \frac {C_{2}}{2}\right ) e^{- 4 t} \cos {\left (t \right )} + \frac {31 e^{t} \sin ^{2}{\left (t \right )}}{26} + \frac {31 e^{t} \cos ^{2}{\left (t \right )}}{26} - \frac {93 \sin ^{2}{\left (t \right )}}{17} - \frac {93 \cos ^{2}{\left (t \right )}}{17}, \ y{\left (t \right )} = C_{1} e^{- 4 t} \cos {\left (t \right )} - C_{2} e^{- 4 t} \sin {\left (t \right )} - \frac {2 e^{t} \sin ^{2}{\left (t \right )}}{13} - \frac {2 e^{t} \cos ^{2}{\left (t \right )}}{13} + \frac {6 \sin ^{2}{\left (t \right )}}{17} + \frac {6 \cos ^{2}{\left (t \right )}}{17}\right ]
\]