52.10.37 problem 40
Internal
problem
ID
[8431]
Book
:
DIFFERENTIAL
EQUATIONS
with
Boundary
Value
Problems.
DENNIS
G.
ZILL,
WARREN
S.
WRIGHT,
MICHAEL
R.
CULLEN.
Brooks/Cole.
Boston,
MA.
2013.
8th
edition.
Section
:
CHAPTER
8
SYSTEMS
OF
LINEAR
FIRST-ORDER
DIFFERENTIAL
EQUATIONS.
EXERCISES
8.2.
Page
346
Problem
number
:
40
Date
solved
:
Sunday, March 30, 2025 at 01:04:42 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 )+2 z \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=3 x \left (t \right )+6 z \left (t \right )\\ \frac {d}{d t}z \left (t \right )&=-4 x \left (t \right )-3 z \left (t \right ) \end{align*}
✓ Maple. Time used: 0.139 (sec). Leaf size: 91
ode:=[diff(x(t),t) = 2*x(t)+y(t)+2*z(t), diff(y(t),t) = 3*x(t)+6*z(t), diff(z(t),t) = -4*x(t)-3*z(t)];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= -\frac {{\mathrm e}^{t} \left (2 \sin \left (2 t \right ) c_2 -\sin \left (2 t \right ) c_3 +\cos \left (2 t \right ) c_2 +2 \cos \left (2 t \right ) c_3 \right )}{2} \\
y \left (t \right ) &= -2 c_1 \,{\mathrm e}^{-3 t}-\frac {3 c_2 \,{\mathrm e}^{t} \cos \left (2 t \right )}{2}+\frac {3 c_3 \,{\mathrm e}^{t} \sin \left (2 t \right )}{2} \\
z \left (t \right ) &= c_1 \,{\mathrm e}^{-3 t}+c_2 \,{\mathrm e}^{t} \sin \left (2 t \right )+c_3 \,{\mathrm e}^{t} \cos \left (2 t \right ) \\
\end{align*}
✓ Mathematica. Time used: 0.013 (sec). Leaf size: 176
ode={D[x[t],t]==2*x[t]+y[t]+2*z[t],D[y[t],t]==3*x[t]+6*z[t],D[z[t],t]==-4*x[t]-3*z[t]};
ic={};
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to \frac {1}{2} e^t (2 c_1 \cos (2 t)+(c_1+c_2+2 c_3) \sin (2 t)) \\
y(t)\to \frac {2}{5} (-3 c_1+c_2-3 c_3) e^{-3 t}+\frac {3}{5} (2 c_1+c_2+2 c_3) e^t \cos (2 t)-\frac {3}{5} (3 c_1-c_2-2 c_3) e^t \sin (t) \cos (t) \\
z(t)\to \frac {1}{5} e^{-3 t} \left (-(3 c_1-c_2-2 c_3) e^{4 t} \cos (2 t)-2 (2 c_1+c_2+2 c_3) e^{4 t} \sin (2 t)+3 c_1-c_2+3 c_3\right ) \\
\end{align*}
✓ Sympy. Time used: 0.184 (sec). Leaf size: 95
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
z = Function("z")
ode=[Eq(-2*x(t) - y(t) - 2*z(t) + Derivative(x(t), t),0),Eq(-3*x(t) - 6*z(t) + Derivative(y(t), t),0),Eq(4*x(t) + 3*z(t) + Derivative(z(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
\[
\left [ x{\left (t \right )} = \left (\frac {C_{1}}{2} + C_{2}\right ) e^{t} \sin {\left (2 t \right )} - \left (C_{1} - \frac {C_{2}}{2}\right ) e^{t} \cos {\left (2 t \right )}, \ y{\left (t \right )} = \frac {3 C_{1} e^{t} \sin {\left (2 t \right )}}{2} + \frac {3 C_{2} e^{t} \cos {\left (2 t \right )}}{2} - 2 C_{3} e^{- 3 t}, \ z{\left (t \right )} = C_{1} e^{t} \cos {\left (2 t \right )} - C_{2} e^{t} \sin {\left (2 t \right )} + C_{3} e^{- 3 t}\right ]
\]