67.7.4 problem Problem 3(d)
Internal
problem
ID
[14036]
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(d)
Date
solved
:
Wednesday, March 05, 2025 at 10:26:39 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=-14 x \left (t \right )+39 y \left (t \right )+78 \sinh \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-6 x \left (t \right )+16 y \left (t \right )+6 \cosh \left (t \right ) \end{align*}
✓ Maple. Time used: 1.367 (sec). Leaf size: 85
ode:=[diff(x(t),t) = -14*x(t)+39*y(t)+78*sinh(t), diff(y(t),t) = -6*x(t)+16*y(t)+6*cosh(t)];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= {\mathrm e}^{t} \sin \left (3 t \right ) c_{2} +{\mathrm e}^{t} \cos \left (3 t \right ) c_{1} +60 \,{\mathrm e}^{-t}-52 \,{\mathrm e}^{t} \\
y &= \frac {5 \,{\mathrm e}^{t} \sin \left (3 t \right ) c_{2}}{13}+\frac {{\mathrm e}^{t} \cos \left (3 t \right ) c_{2}}{13}+\frac {5 \,{\mathrm e}^{t} \cos \left (3 t \right ) c_{1}}{13}-\frac {{\mathrm e}^{t} \sin \left (3 t \right ) c_{1}}{13}+20 \,{\mathrm e}^{-t}-20 \,{\mathrm e}^{t}-2 \sinh \left (t \right ) \\
\end{align*}
✓ Mathematica. Time used: 15.4 (sec). Leaf size: 1975
ode={D[x[t],t]==-14*x[t]+39*y[t]+78*Sinh[t],D[y[t],t]==-6*x[t]+16*y[t]+6*Cosh[t]};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
Too large to display
✓ Sympy. Time used: 0.827 (sec). Leaf size: 129
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(14*x(t) - 39*y(t) - 78*sinh(t) + Derivative(x(t), t),0),Eq(6*x(t) - 16*y(t) - 6*cosh(t) + Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = \left (\frac {C_{1}}{2} + \frac {5 C_{2}}{2}\right ) e^{t} \cos {\left (3 t \right )} - \left (\frac {5 C_{1}}{2} - \frac {C_{2}}{2}\right ) e^{t} \sin {\left (3 t \right )} - 112 \sin ^{2}{\left (3 t \right )} \sinh {\left (t \right )} + 8 \sin ^{2}{\left (3 t \right )} \cosh {\left (t \right )} - 112 \cos ^{2}{\left (3 t \right )} \sinh {\left (t \right )} + 8 \cos ^{2}{\left (3 t \right )} \cosh {\left (t \right )}, \ y{\left (t \right )} = - C_{1} e^{t} \sin {\left (3 t \right )} + C_{2} e^{t} \cos {\left (3 t \right )} - 42 \sin ^{2}{\left (3 t \right )} \sinh {\left (t \right )} - 42 \cos ^{2}{\left (3 t \right )} \sinh {\left (t \right )}\right ]
\]