85.93.1 problem 1
Internal
problem
ID
[23062]
Book
:
Applied
Differential
Equations.
By
Murray
R.
Spiegel.
3rd
edition.
1980.
Pearson.
ISBN
978-0130400970
Section
:
Chapter
11.
Matrix
eigenvalue
methods
for
systems
of
linear
differential
equations.
B
Exercises
at
page
529
Problem
number
:
1
Date
solved
:
Thursday, October 02, 2025 at 09:18:28 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )+4 x \left (t \right )+2 y \left (t \right )-z \left (t \right )&=12 \,{\mathrm e}^{t}\\ \frac {d}{d t}y \left (t \right )-2 x \left (t \right )-5 y \left (t \right )+3 z \left (t \right )&=0\\ \frac {d}{d t}z \left (t \right )+4 x \left (t \right )+z \left (t \right )&=30 \,{\mathrm e}^{-t} \end{align*}
✓ Maple. Time used: 0.128 (sec). Leaf size: 104
ode:=[diff(x(t),t)+4*x(t)+2*y(t)-z(t) = 12*exp(t), diff(y(t),t)-2*x(t)-5*y(t)+3*z(t) = 0, diff(z(t),t)+4*x(t)+z(t) = 30*exp(-t)];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= {\mathrm e}^{4 t} c_3 +{\mathrm e}^{-3 t} c_2 -3 \,{\mathrm e}^{-t}+4 \,{\mathrm e}^{t} \\
y \left (t \right ) &= -\frac {22 \,{\mathrm e}^{4 t} c_3}{5}-8 \,{\mathrm e}^{t}+\frac {{\mathrm e}^{-3 t} c_2}{2}+\frac {{\mathrm e}^{-t} c_1}{2}+21 \,{\mathrm e}^{-t} t +\frac {9 \,{\mathrm e}^{-t}}{2} \\
z \left (t \right ) &= -\frac {4 \,{\mathrm e}^{4 t} c_3}{5}-8 \,{\mathrm e}^{t}+{\mathrm e}^{-t} c_1 +42 \,{\mathrm e}^{-t} t +2 \,{\mathrm e}^{-3 t} c_2 \\
\end{align*}
✓ Mathematica. Time used: 0.454 (sec). Leaf size: 283
ode={D[x[t],{t,1}]+4*x[t]+2*y[t]-z[t]==12*Exp[t], D[y[t],{t,1}]-2*x[t]-5*y[t]+3*z[t]==0,D[z[t],t]+4*x[t]+z[t]==30*Exp[-t]};
ic={};
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
\begin{align*} x(t)&\to \frac {1}{7} e^{-3 t} \left (-42 t-21 e^{2 t}+28 e^{4 t}-42 \left (e^{7 t}-1\right ) \log \left (e^t\right )+e^{7 t} (42 t-c_1-2 c_2+c_3)+8 c_1+2 c_2-c_3\right )\\ y(t)&\to \frac {1}{70} e^{-3 t} \left (-560 e^{4 t}+42 \left (-14 e^{2 t}+44 e^{7 t}+5\right ) \log \left (e^t\right )-5 (42 t-8 c_1-2 c_2+c_3)-44 e^{7 t} (42 t-c_1-2 c_2+c_3)+7 e^{2 t} (294 t+27-12 c_1-4 c_2+7 c_3)\right )\\ z(t)&\to \frac {1}{35} e^{-3 t} \left (-280 e^{4 t}+84 \left (-7 e^{2 t}+2 e^{7 t}+5\right ) \log \left (e^t\right )-10 (42 t-8 c_1-2 c_2+c_3)-4 e^{7 t} (42 t-c_1-2 c_2+c_3)+7 e^{2 t} (294 t-18-12 c_1-4 c_2+7 c_3)\right ) \end{align*}
✓ Sympy. Time used: 0.235 (sec). Leaf size: 109
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
z = Function("z")
ode=[Eq(4*x(t) + 2*y(t) - z(t) - 12*exp(t) + Derivative(x(t), t),0),Eq(-2*x(t) - 5*y(t) + 3*z(t) + Derivative(y(t), t),0),Eq(4*x(t) + z(t) + Derivative(z(t), t) - 30*exp(-t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
\[
\left [ x{\left (t \right )} = \frac {C_{1} e^{- 3 t}}{2} - \frac {5 C_{2} e^{4 t}}{4} + 4 e^{t} - 3 e^{- t}, \ y{\left (t \right )} = \frac {C_{1} e^{- 3 t}}{4} + \frac {11 C_{2} e^{4 t}}{2} + 21 t e^{- t} + \left (\frac {C_{3}}{2} + \frac {27}{10}\right ) e^{- t} - 8 e^{t}, \ z{\left (t \right )} = C_{1} e^{- 3 t} + C_{2} e^{4 t} + 42 t e^{- t} + \left (C_{3} - \frac {18}{5}\right ) e^{- t} - 8 e^{t}\right ]
\]