67.7.6 problem Problem 4(b)
Internal
problem
ID
[14038]
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
4(b)
Date
solved
:
Wednesday, March 05, 2025 at 10:26:44 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=2 x \left (t \right )+6 y \left (t \right )-2 z \left (t \right )+50 \,{\mathrm e}^{t}\\ \frac {d}{d t}y \left (t \right )&=6 x \left (t \right )+2 y \left (t \right )-2 z \left (t \right )+21 \,{\mathrm e}^{-t}\\ \frac {d}{d t}z \left (t \right )&=-x \left (t \right )+6 y \left (t \right )+z \left (t \right )+9 \end{align*}
✓ Maple. Time used: 0.122 (sec). Leaf size: 101
ode:=[diff(x(t),t) = 2*x(t)+6*y(t)-2*z(t)+50*exp(t), diff(y(t),t) = 6*x(t)+2*y(t)-2*z(t)+21*exp(-t), diff(z(t),t) = -x(t)+6*y(t)+z(t)+9];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= \frac {2 c_{2} {\mathrm e}^{3 t}}{5}+12 \,{\mathrm e}^{t}-1-6 \,{\mathrm e}^{-t}+c_{3} {\mathrm e}^{6 t}+{\mathrm e}^{-4 t} c_{1} \\
y &= \frac {2 c_{2} {\mathrm e}^{3 t}}{5}+2 \,{\mathrm e}^{t}-1+{\mathrm e}^{-t}+c_{3} {\mathrm e}^{6 t}-\frac {2 \,{\mathrm e}^{-4 t} c_{1}}{3} \\
z \left (t \right ) &= c_{2} {\mathrm e}^{3 t}-6 \,{\mathrm e}^{-t}+c_{3} {\mathrm e}^{6 t}+37 \,{\mathrm e}^{t}-4+{\mathrm e}^{-4 t} c_{1} \\
\end{align*}
✓ Mathematica. Time used: 0.146 (sec). Leaf size: 213
ode={D[x[t],t]==2*x[t]+6*y[t]-2*z[t]+50*Exp[t],D[y[t],t]==6*x[t]+2*y[t]-2*z[t]+21*Exp[-t],D[z[t],t]==-x[t]+6*y[t]+z[t]+9};
ic={};
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to -6 e^{-t}+12 e^t+\frac {3}{5} (c_1-c_2) e^{-4 t}+\frac {1}{15} (16 c_1+9 c_2-10 c_3) e^{6 t}-\frac {2}{3} (c_1-c_3) e^{3 t}-1 \\
y(t)\to e^{-t}+2 e^t-\frac {2}{5} (c_1-c_2) e^{-4 t}+\frac {1}{15} (16 c_1+9 c_2-10 c_3) e^{6 t}-\frac {2}{3} (c_1-c_3) e^{3 t}-1 \\
z(t)\to -6 e^{-t}+37 e^t+\frac {3}{5} (c_1-c_2) e^{-4 t}+\frac {1}{15} (16 c_1+9 c_2-10 c_3) e^{6 t}-\frac {5}{3} (c_1-c_3) e^{3 t}-4 \\
\end{align*}
✓ Sympy. Time used: 0.433 (sec). Leaf size: 112
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
z = Function("z")
ode=[Eq(-2*x(t) - 6*y(t) + 2*z(t) - 50*exp(t) + Derivative(x(t), t),0),Eq(-6*x(t) - 2*y(t) + 2*z(t) + Derivative(y(t), t) - 21*exp(-t),0),Eq(x(t) - 6*y(t) - z(t) + Derivative(z(t), t) - 9,0)]
ics = {}
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
\[
\left [ x{\left (t \right )} = C_{1} e^{- 4 t} + \frac {2 C_{2} e^{3 t}}{5} + C_{3} e^{6 t} + 12 e^{t} - 1 - 6 e^{- t}, \ y{\left (t \right )} = - \frac {2 C_{1} e^{- 4 t}}{3} + \frac {2 C_{2} e^{3 t}}{5} + C_{3} e^{6 t} + 2 e^{t} - 1 + e^{- t}, \ z{\left (t \right )} = C_{1} e^{- 4 t} + C_{2} e^{3 t} + C_{3} e^{6 t} + 37 e^{t} - 4 - 6 e^{- t}\right ]
\]