7.23.6 problem 6

Internal problem ID [592]
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 : 6
Date solved : Tuesday, March 04, 2025 at 11:27:21 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=x \left (t \right )+9 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-2 x \left (t \right )-5 y \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x \left (0\right ) = 3\\ y \left (0\right ) = 2 \end{align*}

✓ Maple. Time used: 0.023 (sec). Leaf size: 44

ode:=[diff(x(t),t) = x(t)+9*y(t), diff(y(t),t) = -2*x(t)-5*y(t)]; 
ic:=x(0) = 3y(0) = 2; 
dsolve([ode,ic]);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-2 t} \left (9 \sin \left (3 t \right )+3 \cos \left (3 t \right )\right ) \\ y \left (t \right ) &= -\frac {{\mathrm e}^{-2 t} \left (12 \sin \left (3 t \right )-6 \cos \left (3 t \right )\right )}{3} \\ \end{align*}

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 44

ode={D[x[t],t]==x[t]+9*y[t],D[y[t],t]==-2*x[t]-5*y[t]}; 
ic={x[0]==3,y[0]==2}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to 3 e^{-2 t} (3 \sin (3 t)+\cos (3 t)) \\ y(t)\to 2 e^{-2 t} (\cos (3 t)-2 \sin (3 t)) \\ \end{align*}

✓ Sympy. Time used: 0.130 (sec). Leaf size: 68

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-x(t) - 9*y(t) + Derivative(x(t), t),0),Eq(2*x(t) + 5*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - \left (\frac {3 C_{1}}{2} - \frac {3 C_{2}}{2}\right ) e^{- 2 t} \cos {\left (3 t \right )} + \left (\frac {3 C_{1}}{2} + \frac {3 C_{2}}{2}\right ) e^{- 2 t} \sin {\left (3 t \right )}, \ y{\left (t \right )} = C_{1} e^{- 2 t} \cos {\left (3 t \right )} - C_{2} e^{- 2 t} \sin {\left (3 t \right )}\right ] \]