61.7.15 problem Problem 6(c)
Internal
problem
ID
[15407]
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(c)
Date
solved
:
Thursday, October 02, 2025 at 10:13:42 AM
CAS
classification
:
system_of_ODEs
\begin{align*} x^{\prime }\left (t \right )&=-3 x \left (t \right )+3 y+z \left (t \right )+5 \sin \left (2 t \right )\\ y^{\prime }&=x \left (t \right )-5 y-3 z \left (t \right )+5 \cos \left (2 t \right )\\ z^{\prime }\left (t \right )&=-3 x \left (t \right )+7 y+3 z \left (t \right )+23 \,{\mathrm e}^{t} \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: 1.220 (sec). Leaf size: 131
ode:=[diff(x(t),t) = -3*x(t)+3*y(t)+z(t)+5*sin(2*t), diff(y(t),t) = x(t)-5*y(t)-3*z(t)+5*cos(2*t), diff(z(t),t) = -3*x(t)+7*y(t)+3*z(t)+23*exp(t)];
ic:=[x(0) = 1, y(0) = 2, z(0) = 3];
dsolve([ode,op(ic)]);
\begin{align*}
x \left (t \right ) &= -\frac {69 \,{\mathrm e}^{t}}{26}+\sin \left (2 t \right )+\frac {\cos \left (2 t \right )}{2}+\frac {21 \,{\mathrm e}^{-t}}{2}-\frac {191 \,{\mathrm e}^{-2 t} \cos \left (2 t \right )}{26}+\frac {16 \,{\mathrm e}^{-2 t} \sin \left (2 t \right )}{13} \\
y \left (t \right ) &= -\frac {253 \,{\mathrm e}^{t}}{26}-\frac {5 \sin \left (2 t \right )}{2}+\frac {21 \,{\mathrm e}^{-t}}{2}+\frac {191 \,{\mathrm e}^{-2 t} \sin \left (2 t \right )}{26}+\frac {16 \,{\mathrm e}^{-2 t} \cos \left (2 t \right )}{13} \\
z \left (t \right ) &= \frac {483 \,{\mathrm e}^{t}}{26}+\frac {7 \cos \left (2 t \right )}{2}+\frac {9 \sin \left (2 t \right )}{2}-\frac {21 \,{\mathrm e}^{-t}}{2}-\frac {223 \,{\mathrm e}^{-2 t} \cos \left (2 t \right )}{26}-\frac {159 \,{\mathrm e}^{-2 t} \sin \left (2 t \right )}{26} \\
\end{align*}
✓ Mathematica. Time used: 5.232 (sec). Leaf size: 3703
ode={D[x[t],t]==-3*x[t]+3*y[t]+z[t]+5*Sin[3*t],D[y[t],t]==x[t]-5*y[t]-3*z[t]+5*Cos[2*t],D[z[t],t]==-3*x[t]+7*y[t]+3*z[t]+23*Exp[t]};
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: 0.776 (sec). Leaf size: 393
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
z = Function("z")
ode=[Eq(3*x(t) - 3*y(t) - z(t) - 5*sin(2*t) + Derivative(x(t), t),0),Eq(-x(t) + 5*y(t) + 3*z(t) - 5*cos(2*t) + Derivative(y(t), t),0),Eq(3*x(t) - 7*y(t) - 3*z(t) - 23*exp(t) + Derivative(z(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
\[
\left [ x{\left (t \right )} = - C_{1} e^{- t} + \left (\frac {C_{2}}{2} - \frac {C_{3}}{2}\right ) e^{- 2 t} \cos {\left (2 t \right )} - \left (\frac {C_{2}}{2} + \frac {C_{3}}{2}\right ) e^{- 2 t} \sin {\left (2 t \right )} + \frac {115 e^{t} \sin ^{2}{\left (2 t \right )}}{13} + \frac {115 e^{t} \cos ^{2}{\left (2 t \right )}}{13} - \frac {23 e^{t}}{2} + 2 \sin ^{3}{\left (2 t \right )} + \frac {7 \sin ^{2}{\left (2 t \right )} \cos {\left (2 t \right )}}{2} + 2 \sin {\left (2 t \right )} \cos ^{2}{\left (2 t \right )} - \sin {\left (2 t \right )} + \frac {7 \cos ^{3}{\left (2 t \right )}}{2} - 3 \cos {\left (2 t \right )}, \ y{\left (t \right )} = - C_{1} e^{- t} - \left (\frac {C_{2}}{2} - \frac {C_{3}}{2}\right ) e^{- 2 t} \sin {\left (2 t \right )} - \left (\frac {C_{2}}{2} + \frac {C_{3}}{2}\right ) e^{- 2 t} \cos {\left (2 t \right )} + \frac {23 e^{t} \sin ^{2}{\left (2 t \right )}}{13} + \frac {23 e^{t} \cos ^{2}{\left (2 t \right )}}{13} - \frac {23 e^{t}}{2} - \frac {3 \sin ^{3}{\left (2 t \right )}}{2} + 3 \sin ^{2}{\left (2 t \right )} \cos {\left (2 t \right )} - \frac {3 \sin {\left (2 t \right )} \cos ^{2}{\left (2 t \right )}}{2} - \sin {\left (2 t \right )} + 3 \cos ^{3}{\left (2 t \right )} - 3 \cos {\left (2 t \right )}, \ z{\left (t \right )} = C_{1} e^{- t} + C_{2} e^{- 2 t} \cos {\left (2 t \right )} - C_{3} e^{- 2 t} \sin {\left (2 t \right )} + \frac {92 e^{t} \sin ^{2}{\left (2 t \right )}}{13} + \frac {92 e^{t} \cos ^{2}{\left (2 t \right )}}{13} + \frac {23 e^{t}}{2} + \frac {7 \sin ^{3}{\left (2 t \right )}}{2} + \frac {\sin ^{2}{\left (2 t \right )} \cos {\left (2 t \right )}}{2} + \frac {7 \sin {\left (2 t \right )} \cos ^{2}{\left (2 t \right )}}{2} + \sin {\left (2 t \right )} + \frac {\cos ^{3}{\left (2 t \right )}}{2} + 3 \cos {\left (2 t \right )}\right ]
\]