67.7.14 problem Problem 6(b)
Internal
problem
ID
[14054]
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
6(b)
Date
solved
:
Monday, March 31, 2025 at 12:02:02 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=2 x \left (t \right )+y \left (t \right )-z \left (t \right )+5 \sin \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=y \left (t \right )+z \left (t \right )-10 \cos \left (t \right )\\ \frac {d}{d t}z \left (t \right )&=x \left (t \right )+z \left (t \right )+2 \end{align*}
With initial conditions
\begin{align*} x \left (0\right ) = 1\\ y \left (0\right ) = 2\\ z \left (0\right ) = 3 \end{align*}
✓ Maple. Time used: 0.474 (sec). Leaf size: 70
ode:=[diff(x(t),t) = 2*x(t)+y(t)-z(t)+5*sin(t), diff(y(t),t) = y(t)+z(t)-10*cos(t), diff(z(t),t) = x(t)+z(t)+2];
ic:=x(0) = 1y(0) = 2z(0) = 3;
dsolve([ode,ic]);
\begin{align*}
x \left (t \right ) &= -2 \cos \left (t \right )-1-3 \,{\mathrm e}^{t} \sin \left (t \right )+4 \,{\mathrm e}^{t} \cos \left (t \right ) \\
y \left (t \right ) &= 5 \cos \left (t \right )-4 \sin \left (t \right )+1+3 \,{\mathrm e}^{t} \sin \left (t \right )-4 \,{\mathrm e}^{t} \cos \left (t \right ) \\
z \left (t \right ) &= -1+\cos \left (t \right )-\sin \left (t \right )+3 \,{\mathrm e}^{t} \cos \left (t \right )+4 \,{\mathrm e}^{t} \sin \left (t \right ) \\
\end{align*}
✓ Mathematica. Time used: 1.098 (sec). Leaf size: 2170
ode={D[x[t],t]==2*x[t]+y[t]-z[t]+5*Sin[t],D[y[t],t]==y[t]+z[t]-10*Cos[t],D[z[t],t]==x[t]+z[t]+2};
ic={x[0]==1,y[0]==2,z[0]==3};
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
Too large to display
✓ Sympy. Time used: 0.628 (sec). Leaf size: 248
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
z = Function("z")
ode=[Eq(-2*x(t) - y(t) + z(t) - 5*sin(t) + Derivative(x(t), t),0),Eq(-y(t) - z(t) + 10*cos(t) + Derivative(y(t), t),0),Eq(-x(t) - z(t) + Derivative(z(t), t) - 2,0)]
ics = {}
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
\[
\left [ x{\left (t \right )} = - C_{1} e^{t} \sin {\left (t \right )} - C_{2} e^{t} \cos {\left (t \right )} + C_{3} e^{2 t} + 2 \sin ^{3}{\left (t \right )} - \frac {7 \sin ^{2}{\left (t \right )} \cos {\left (t \right )}}{2} - \sin ^{2}{\left (t \right )} + 2 \sin {\left (t \right )} \cos ^{2}{\left (t \right )} - 2 \sin {\left (t \right )} - \frac {7 \cos ^{3}{\left (t \right )}}{2} - \cos ^{2}{\left (t \right )} + \frac {3 \cos {\left (t \right )}}{2}, \ y{\left (t \right )} = C_{1} e^{t} \sin {\left (t \right )} + C_{2} e^{t} \cos {\left (t \right )} + C_{3} e^{2 t} - 2 \sin ^{3}{\left (t \right )} + \frac {7 \sin ^{2}{\left (t \right )} \cos {\left (t \right )}}{2} + \sin ^{2}{\left (t \right )} - 2 \sin {\left (t \right )} \cos ^{2}{\left (t \right )} - 2 \sin {\left (t \right )} + \frac {7 \cos ^{3}{\left (t \right )}}{2} + \cos ^{2}{\left (t \right )} + \frac {3 \cos {\left (t \right )}}{2}, \ z{\left (t \right )} = C_{1} e^{t} \cos {\left (t \right )} - C_{2} e^{t} \sin {\left (t \right )} + C_{3} e^{2 t} + \sin ^{3}{\left (t \right )} - \frac {\sin ^{2}{\left (t \right )} \cos {\left (t \right )}}{2} - \sin ^{2}{\left (t \right )} + \sin {\left (t \right )} \cos ^{2}{\left (t \right )} - 2 \sin {\left (t \right )} - \frac {\cos ^{3}{\left (t \right )}}{2} - \cos ^{2}{\left (t \right )} + \frac {3 \cos {\left (t \right )}}{2}\right ]
\]