7.23.13 problem 19
Internal
problem
ID
[599]
Book
:
Elementary
Differential
Equations.
By
C.
Henry
Edwards,
David
E.
Penney
and
David
Calvis.
6th
edition.
2008
Section
:
Chapter
5.
Linear
systems
of
differential
equations.
Section
5.2
(Applications).
Problems
at
page
345
Problem
number
:
19
Date
solved
:
Saturday, March 29, 2025 at 04:57:39 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=x \left (t \right )-2 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-4 x \left (t \right )+4 y \left (t \right )-2 z \left (t \right )\\ \frac {d}{d t}z \left (t \right )&=-4 y \left (t \right )+4 z \left (t \right ) \end{align*}
✓ Maple. Time used: 0.214 (sec). Leaf size: 910
ode:=[diff(x(t),t) = x(t)-2*y(t), diff(y(t),t) = -4*x(t)+4*y(t)-2*z(t), diff(z(t),t) = -4*y(t)+4*z(t)];
dsolve(ode);
\begin{align*} \text {Solution too large to show}\end{align*}
✓ Mathematica. Time used: 0.019 (sec). Leaf size: 527
ode={D[x[t],t]==x[t]-2*y[t],D[y[t],t]==-4*x[t]+4*y[t]-2*z[t],D[z[t],t]==-4*y[t]+4*z[t]};
ic={};
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to 4 c_3 \text {RootSum}\left [\text {$\#$1}^3-9 \text {$\#$1}^2+8 \text {$\#$1}+24\&,\frac {e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-18 \text {$\#$1}+8}\&\right ]-2 c_2 \text {RootSum}\left [\text {$\#$1}^3-9 \text {$\#$1}^2+8 \text {$\#$1}+24\&,\frac {\text {$\#$1} e^{\text {$\#$1} t}-4 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-18 \text {$\#$1}+8}\&\right ]+c_1 \text {RootSum}\left [\text {$\#$1}^3-9 \text {$\#$1}^2+8 \text {$\#$1}+24\&,\frac {\text {$\#$1}^2 e^{\text {$\#$1} t}-8 \text {$\#$1} e^{\text {$\#$1} t}+8 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-18 \text {$\#$1}+8}\&\right ] \\
y(t)\to -4 c_1 \text {RootSum}\left [\text {$\#$1}^3-9 \text {$\#$1}^2+8 \text {$\#$1}+24\&,\frac {\text {$\#$1} e^{\text {$\#$1} t}-4 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-18 \text {$\#$1}+8}\&\right ]-2 c_3 \text {RootSum}\left [\text {$\#$1}^3-9 \text {$\#$1}^2+8 \text {$\#$1}+24\&,\frac {\text {$\#$1} e^{\text {$\#$1} t}-e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-18 \text {$\#$1}+8}\&\right ]+c_2 \text {RootSum}\left [\text {$\#$1}^3-9 \text {$\#$1}^2+8 \text {$\#$1}+24\&,\frac {\text {$\#$1}^2 e^{\text {$\#$1} t}-5 \text {$\#$1} e^{\text {$\#$1} t}+4 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-18 \text {$\#$1}+8}\&\right ] \\
z(t)\to 16 c_1 \text {RootSum}\left [\text {$\#$1}^3-9 \text {$\#$1}^2+8 \text {$\#$1}+24\&,\frac {e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-18 \text {$\#$1}+8}\&\right ]-4 c_2 \text {RootSum}\left [\text {$\#$1}^3-9 \text {$\#$1}^2+8 \text {$\#$1}+24\&,\frac {\text {$\#$1} e^{\text {$\#$1} t}-e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-18 \text {$\#$1}+8}\&\right ]+c_3 \text {RootSum}\left [\text {$\#$1}^3-9 \text {$\#$1}^2+8 \text {$\#$1}+24\&,\frac {\text {$\#$1}^2 e^{\text {$\#$1} t}-5 \text {$\#$1} e^{\text {$\#$1} t}-4 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-18 \text {$\#$1}+8}\&\right ] \\
\end{align*}
✓ Sympy. Time used: 14.988 (sec). Leaf size: 1574
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
z = Function("z")
ode=[Eq(-x(t) + 2*y(t) + Derivative(x(t), t),0),Eq(4*x(t) - 4*y(t) + 2*z(t) + Derivative(y(t), t),0),Eq(4*y(t) - 4*z(t) + Derivative(z(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
\[
\text {Solution too large to show}
\]