85.91.3 problem 8 (c)
Internal
problem
ID
[23049]
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.
A
Exercises
at
page
509
Problem
number
:
8
(c)
Date
solved
:
Thursday, October 02, 2025 at 09:18:18 PM
CAS
classification
:
system_of_ODEs
\begin{align*} z \left (t \right )+\frac {d}{d t}x \left (t \right )&=x \left (t \right )\\ \frac {d}{d t}y \left (t \right )-2 x \left (t \right )&=y \left (t \right )+3 t\\ \frac {d}{d t}z \left (t \right )+4 y \left (t \right )&=z \left (t \right )-\cos \left (t \right ) \end{align*}
✓ Maple. Time used: 0.695 (sec). Leaf size: 268
ode:=[diff(x(t),t)+z(t) = x(t), diff(y(t),t)-2*x(t) = y(t)+3*t, diff(z(t),t)+4*y(t) = z(t)-cos(t)];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= -\frac {8}{9 \left (\sqrt {3}-1\right ) \left (1+\sqrt {3}\right )}+\frac {30 c_1 \,{\mathrm e}^{3 t}-40 t +30 c_2 \cos \left (\sqrt {3}\, t \right )+30 c_3 \sin \left (\sqrt {3}\, t \right )+6 \cos \left (t \right )+3 \sin \left (t \right )}{15 \left (\sqrt {3}-1\right ) \left (1+\sqrt {3}\right )} \\
y \left (t \right ) &= \frac {10}{9 \left (\sqrt {3}-1\right ) \left (1+\sqrt {3}\right )}-\frac {15 c_3 \sqrt {3}\, \cos \left (\sqrt {3}\, t \right )-15 c_2 \sqrt {3}\, \sin \left (\sqrt {3}\, t \right )-30 c_1 \,{\mathrm e}^{3 t}+15 c_2 \cos \left (\sqrt {3}\, t \right )+15 c_3 \sin \left (\sqrt {3}\, t \right )+9 \cos \left (t \right )-3 \sin \left (t \right )+10 t}{15 \left (\sqrt {3}-1\right ) \left (1+\sqrt {3}\right )} \\
z \left (t \right ) &= \frac {16}{9 \left (\sqrt {3}-1\right ) \left (1+\sqrt {3}\right )}+\frac {-60 c_1 \,{\mathrm e}^{3 t}-40 t -30 c_3 \sqrt {3}\, \cos \left (\sqrt {3}\, t \right )+30 c_2 \sqrt {3}\, \sin \left (\sqrt {3}\, t \right )+30 c_2 \cos \left (\sqrt {3}\, t \right )+30 c_3 \sin \left (\sqrt {3}\, t \right )+3 \cos \left (t \right )+9 \sin \left (t \right )}{15 \left (\sqrt {3}-1\right ) \left (1+\sqrt {3}\right )} \\
\end{align*}
✓ Mathematica. Time used: 2.045 (sec). Leaf size: 304
ode={D[x[t],{t,1}]+z[t]==x[t], D[y[t],{t,1}]-2*x[t]==y[t]+3*t, D[z[t],t]+4*y[t]==z[t]-Cos[t]};
ic={};
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
\begin{align*} x(t)&\to \frac {1}{90} \left (-120 t+9 \sin (t)+18 \cos (t)+30 c_1 e^{3 t}+30 c_2 e^{3 t}-15 c_3 e^{3 t}+15 (4 c_1-2 c_2+c_3) \cos \left (\sqrt {3} t\right )-30 \sqrt {3} c_2 \sin \left (\sqrt {3} t\right )-15 \sqrt {3} c_3 \sin \left (\sqrt {3} t\right )-40\right )\\ y(t)&\to \frac {1}{90} \left (-30 t+9 \sin (t)-27 \cos (t)+30 c_1 e^{3 t}+30 c_2 e^{3 t}-15 c_3 e^{3 t}+15 (-2 c_1+4 c_2+c_3) \cos \left (\sqrt {3} t\right )+30 \sqrt {3} c_1 \sin \left (\sqrt {3} t\right )+15 \sqrt {3} c_3 \sin \left (\sqrt {3} t\right )+50\right )\\ z(t)&\to \frac {1}{90} \left (-120 t+27 \sin (t)+9 \cos (t)-60 c_1 e^{3 t}-60 c_2 e^{3 t}+30 c_3 e^{3 t}+60 (c_1+c_2+c_3) \cos \left (\sqrt {3} t\right )+60 \sqrt {3} c_1 \sin \left (\sqrt {3} t\right )-60 \sqrt {3} c_2 \sin \left (\sqrt {3} t\right )+80\right ) \end{align*}
✓ Sympy. Time used: 0.547 (sec). Leaf size: 466
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
z = Function("z")
ode=[Eq(-x(t) + z(t) + Derivative(x(t), t),0),Eq(-3*t - 2*x(t) - y(t) + Derivative(y(t), t),0),Eq(4*y(t) - z(t) + cos(t) + Derivative(z(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
\[
\text {Solution too large to show}
\]