73.27.17 problem 38.10 (k)
Internal
problem
ID
[15702]
Book
:
Ordinary
Differential
Equations.
An
introduction
to
the
fundamentals.
Kenneth
B.
Howell.
second
edition.
CRC
Press.
FL,
USA.
2020
Section
:
Chapter
38.
Systems
of
differential
equations.
A
starting
point.
Additional
Exercises.
page
786
Problem
number
:
38.10
(k)
Date
solved
:
Monday, March 31, 2025 at 01:45:29 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=4 x \left (t \right )-13 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right )+19 \cos \left (4 t \right )-13 \sin \left (4 t \right ) \end{align*}
With initial conditions
\begin{align*} x \left (0\right ) = 13\\ y \left (0\right ) = 0 \end{align*}
✓ Maple. Time used: 0.589 (sec). Leaf size: 26
ode:=[diff(x(t),t) = 4*x(t)-13*y(t), diff(y(t),t) = x(t)+19*cos(4*t)-13*sin(4*t)];
ic:=x(0) = 13y(0) = 0;
dsolve([ode,ic]);
\begin{align*}
x \left (t \right ) &= 13 \cos \left (4 t \right )+13 \sin \left (4 t \right ) \\
y \left (t \right ) &= 8 \sin \left (4 t \right ) \\
\end{align*}
✓ Mathematica. Time used: 0.319 (sec). Leaf size: 554
ode={D[x[t],t]==4*x[t]-13*y[t],D[y[t],t]==x[t]+19*Cos[4*t]-13*Sin[4*t]};
ic={x[0]==13,y[0]==0};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to \frac {1}{3} e^{2 t} \left (-\left ((2 \sin (3 t)+3 \cos (3 t)) \int _1^0\frac {13}{3} e^{-2 K[1]} \sin (3 K[1]) (19 \cos (4 K[1])-13 \sin (4 K[1]))dK[1]\right )+(2 \sin (3 t)+3 \cos (3 t)) \int _1^t\frac {13}{3} e^{-2 K[1]} \sin (3 K[1]) (19 \cos (4 K[1])-13 \sin (4 K[1]))dK[1]+13 \left (\sin (3 t) \int _1^0\frac {1}{3} e^{-2 K[2]} (3 \cos (3 K[2])+2 \sin (3 K[2])) (19 \cos (4 K[2])-13 \sin (4 K[2]))dK[2]-\sin (3 t) \int _1^t\frac {1}{3} e^{-2 K[2]} (3 \cos (3 K[2])+2 \sin (3 K[2])) (19 \cos (4 K[2])-13 \sin (4 K[2]))dK[2]+2 \sin (3 t)+3 \cos (3 t)\right )\right ) \\
y(t)\to \frac {1}{3} e^{2 t} \left (\sin (3 t) \left (-\int _1^0\frac {13}{3} e^{-2 K[1]} \sin (3 K[1]) (19 \cos (4 K[1])-13 \sin (4 K[1]))dK[1]\right )+\sin (3 t) \int _1^t\frac {13}{3} e^{-2 K[1]} \sin (3 K[1]) (19 \cos (4 K[1])-13 \sin (4 K[1]))dK[1]+2 \sin (3 t) \int _1^0\frac {1}{3} e^{-2 K[2]} (3 \cos (3 K[2])+2 \sin (3 K[2])) (19 \cos (4 K[2])-13 \sin (4 K[2]))dK[2]-2 \sin (3 t) \int _1^t\frac {1}{3} e^{-2 K[2]} (3 \cos (3 K[2])+2 \sin (3 K[2])) (19 \cos (4 K[2])-13 \sin (4 K[2]))dK[2]-3 \cos (3 t) \int _1^0\frac {1}{3} e^{-2 K[2]} (3 \cos (3 K[2])+2 \sin (3 K[2])) (19 \cos (4 K[2])-13 \sin (4 K[2]))dK[2]+3 \cos (3 t) \int _1^t\frac {1}{3} e^{-2 K[2]} (3 \cos (3 K[2])+2 \sin (3 K[2])) (19 \cos (4 K[2])-13 \sin (4 K[2]))dK[2]+13 \sin (3 t)\right ) \\
\end{align*}
✓ Sympy. Time used: 0.927 (sec). Leaf size: 143
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(-4*x(t) + 13*y(t) + Derivative(x(t), t),0),Eq(-x(t) + 13*sin(4*t) - 19*cos(4*t) + Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = \left (2 C_{1} - 3 C_{2}\right ) e^{2 t} \cos {\left (3 t \right )} - \left (3 C_{1} + 2 C_{2}\right ) e^{2 t} \sin {\left (3 t \right )} + 13 \sin ^{2}{\left (3 t \right )} \sin {\left (4 t \right )} + 13 \sin ^{2}{\left (3 t \right )} \cos {\left (4 t \right )} + 13 \sin {\left (4 t \right )} \cos ^{2}{\left (3 t \right )} + 13 \cos ^{2}{\left (3 t \right )} \cos {\left (4 t \right )}, \ y{\left (t \right )} = C_{1} e^{2 t} \cos {\left (3 t \right )} - C_{2} e^{2 t} \sin {\left (3 t \right )} + 8 \sin ^{2}{\left (3 t \right )} \sin {\left (4 t \right )} + 8 \sin {\left (4 t \right )} \cos ^{2}{\left (3 t \right )}\right ]
\]