7.22.9 problem 19

Internal problem ID [584]
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 5.1 (First order systems and applications). Problems at page 335
Problem number : 19
Date solved : Tuesday, March 04, 2025 at 11:27:13 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=13 x \left (t \right )+4 y \left (t \right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.020 (sec). Leaf size: 36

ode:=[diff(x(t),t) = -y(t), diff(y(t),t) = 13*x(t)+4*y(t)]; 
ic:=x(0) = 0y(0) = 3; 
dsolve([ode,ic]);
 

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

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 38

ode={D[x[t],t]==-y[t],D[y[t],t]==13*x[t]+4*y[t]}; 
ic={x[0]==0,y[0]==3}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.122 (sec). Leaf size: 70

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(y(t) + Derivative(x(t), t),0),Eq(-13*x(t) - 4*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - \left (\frac {2 C_{1}}{13} + \frac {3 C_{2}}{13}\right ) e^{2 t} \cos {\left (3 t \right )} - \left (\frac {3 C_{1}}{13} - \frac {2 C_{2}}{13}\right ) e^{2 t} \sin {\left (3 t \right )}, \ y{\left (t \right )} = C_{1} e^{2 t} \cos {\left (3 t \right )} - C_{2} e^{2 t} \sin {\left (3 t \right )}\right ] \]