67.8.1 problem Problem 1(a)
Internal
problem
ID
[14049]
Book
:
APPLIED
DIFFERENTIAL
EQUATIONS
The
Primary
Course
by
Vladimir
A.
Dobrushkin.
CRC
Press
2015
Section
:
Chapter
8.4
Systems
of
Linear
Differential
Equations
(Method
of
Undetermined
Coefficients).
Problems
page
520
Problem
number
:
Problem
1(a)
Date
solved
:
Wednesday, March 05, 2025 at 10:27:33 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=x \left (t \right )+5 y \left (t \right )+10 \sinh \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=19 x \left (t \right )-13 y \left (t \right )+24 \sinh \left (t \right ) \end{align*}
✓ Maple. Time used: 1.106 (sec). Leaf size: 135
ode:=[diff(x(t),t) = x(t)+5*y(t)+10*sinh(t), diff(y(t),t) = 19*x(t)-13*y(t)+24*sinh(t)];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= c_{2} {\mathrm e}^{6 t}+{\mathrm e}^{-18 t} c_{1} +\frac {5 \,{\mathrm e}^{-18 t} \left (\left (-\frac {221 \cosh \left (5 t \right )}{60}+\frac {17 \cosh \left (7 t \right )}{7}+\frac {221 \sinh \left (5 t \right )}{60}-\frac {17 \sinh \left (7 t \right )}{7}\right ) {\mathrm e}^{24 t}+\sinh \left (17 t \right )-\frac {221 \sinh \left (19 t \right )}{228}+\cosh \left (17 t \right )-\frac {221 \cosh \left (19 t \right )}{228}\right )}{17} \\
y &= -\frac {2 \cosh \left (7 t \right ) {\mathrm e}^{6 t}}{7}+\frac {2 \sinh \left (7 t \right ) {\mathrm e}^{6 t}}{7}+c_{2} {\mathrm e}^{6 t}-\frac {2 \,{\mathrm e}^{-18 t} \sinh \left (17 t \right )}{17}-\frac {2 \,{\mathrm e}^{-18 t} \cosh \left (17 t \right )}{17}-\frac {19 \,{\mathrm e}^{-18 t} c_{1}}{5}-2 \sinh \left (t \right ) \\
\end{align*}
✓ Mathematica. Time used: 0.044 (sec). Leaf size: 108
ode={D[x[t],t]==x[t]+5*y[t]+10*Sinh[t],D[y[t],t]==19*x[t]-13*y[t]+24*Sinh[t]};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to \frac {120 e^{-t}}{119}-\frac {26 e^t}{19}+\frac {5}{24} (c_1-c_2) e^{-18 t}+\frac {1}{24} (19 c_1+5 c_2) e^{6 t} \\
y(t)\to \frac {71 e^{-t}}{119}-e^t-\frac {19}{24} (c_1-c_2) e^{-18 t}+\frac {1}{24} (19 c_1+5 c_2) e^{6 t} \\
\end{align*}
✓ Sympy. Time used: 0.359 (sec). Leaf size: 61
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(-x(t) - 5*y(t) - 10*sinh(t) + Derivative(x(t), t),0),Eq(-19*x(t) + 13*y(t) - 24*sinh(t) + Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = - \frac {5 C_{1} e^{- 18 t}}{19} + C_{2} e^{6 t} - \frac {5374 \sinh {\left (t \right )}}{2261} - \frac {814 \cosh {\left (t \right )}}{2261}, \ y{\left (t \right )} = C_{1} e^{- 18 t} + C_{2} e^{6 t} - \frac {190 \sinh {\left (t \right )}}{119} - \frac {48 \cosh {\left (t \right )}}{119}\right ]
\]