72.15.3 problem 7
Internal
problem
ID
[14813]
Book
:
DIFFERENTIAL
EQUATIONS
by
Paul
Blanchard,
Robert
L.
Devaney,
Glen
R.
Hall.
4th
edition.
Brooks/Cole.
Boston,
USA.
2012
Section
:
Chapter
3.
Linear
Systems.
Review
Exercises
for
chapter
3.
page
376
Problem
number
:
7
Date
solved
:
Monday, March 31, 2025 at 12:58:57 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=\pi ^{2} x \left (t \right )+\frac {187 y \left (t \right )}{5}\\ \frac {d}{d t}y \left (t \right )&=\sqrt {555}\, x \left (t \right )+\frac {400617 y \left (t \right )}{5000} \end{align*}
With initial conditions
\begin{align*} x \left (0\right ) = 0\\ y \left (0\right ) = 0 \end{align*}
✓ Maple. Time used: 0.027 (sec). Leaf size: 9
ode:=[diff(x(t),t) = Pi^2*x(t)+187/5*y(t), diff(y(t),t) = 555^(1/2)*x(t)+400617/5000*y(t)];
ic:=x(0) = 0y(0) = 0;
dsolve([ode,ic]);
\begin{align*}
x \left (t \right ) &= 0 \\
y \left (t \right ) &= 0 \\
\end{align*}
✓ Mathematica. Time used: 0.026 (sec). Leaf size: 10
ode={D[x[t],t]==Pi^2*x[t]+374/10*y[t],D[y[t],t]==Sqrt[555]*x[t]+801234/10000*y[t]};
ic={x[0]==0,y[0]==0};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to 0 \\
y(t)\to 0 \\
\end{align*}
✓ Sympy. Time used: 0.325 (sec). Leaf size: 218
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(-pi**2*x(t) - 187*y(t)/5 + Derivative(x(t), t),0),Eq(-sqrt(555)*x(t) - 400617*y(t)/5000 + Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = \frac {\sqrt {555} C_{1} \left (-400617 + 5000 \pi ^{2} + \sqrt {- 4006170000 \pi ^{2} + 25000000 \pi ^{4} + 3740000000 \sqrt {555} + 160493980689}\right ) e^{\frac {t \left (5000 \pi ^{2} + 400617 + \sqrt {- 4006170000 \pi ^{2} + 25000000 \pi ^{4} + 3740000000 \sqrt {555} + 160493980689}\right )}{10000}}}{5550000} - \frac {\sqrt {555} C_{2} \left (- 5000 \pi ^{2} + 400617 + \sqrt {- 4006170000 \pi ^{2} + 25000000 \pi ^{4} + 3740000000 \sqrt {555} + 160493980689}\right ) e^{\frac {t \left (- \sqrt {- 4006170000 \pi ^{2} + 25000000 \pi ^{4} + 3740000000 \sqrt {555} + 160493980689} + 5000 \pi ^{2} + 400617\right )}{10000}}}{5550000}, \ y{\left (t \right )} = C_{1} e^{\frac {t \left (5000 \pi ^{2} + 400617 + \sqrt {- 4006170000 \pi ^{2} + 25000000 \pi ^{4} + 3740000000 \sqrt {555} + 160493980689}\right )}{10000}} + C_{2} e^{\frac {t \left (- \sqrt {- 4006170000 \pi ^{2} + 25000000 \pi ^{4} + 3740000000 \sqrt {555} + 160493980689} + 5000 \pi ^{2} + 400617\right )}{10000}}\right ]
\]