72.15.3 problem 7

Internal problem ID [14805]
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 : Thursday, March 13, 2025 at 04:19:16 AM
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: 0.003 (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 &= 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)
 

\[ \left [ x{\left (t \right )} = \frac {\sqrt {555} C_{1} \left (-400617 + 5000 \pi ^{2} + \sqrt {- 4006170000 \pi ^{2} + 25000000 \pi ^{4} + 3740000000 \sqrt {555} + 160493980689}\right ) e^{\frac {t \left (5000 \pi ^{2} + 400617 + \sqrt {- 4006170000 \pi ^{2} + 25000000 \pi ^{4} + 3740000000 \sqrt {555} + 160493980689}\right )}{10000}}}{5550000} - \frac {\sqrt {555} C_{2} \left (- 5000 \pi ^{2} + 400617 + \sqrt {- 4006170000 \pi ^{2} + 25000000 \pi ^{4} + 3740000000 \sqrt {555} + 160493980689}\right ) e^{\frac {t \left (- \sqrt {- 4006170000 \pi ^{2} + 25000000 \pi ^{4} + 3740000000 \sqrt {555} + 160493980689} + 5000 \pi ^{2} + 400617\right )}{10000}}}{5550000}, \ y{\left (t \right )} = C_{1} e^{\frac {t \left (5000 \pi ^{2} + 400617 + \sqrt {- 4006170000 \pi ^{2} + 25000000 \pi ^{4} + 3740000000 \sqrt {555} + 160493980689}\right )}{10000}} + C_{2} e^{\frac {t \left (- \sqrt {- 4006170000 \pi ^{2} + 25000000 \pi ^{4} + 3740000000 \sqrt {555} + 160493980689} + 5000 \pi ^{2} + 400617\right )}{10000}}\right ] \]