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 : Wednesday, February 05, 2025 at 03:45:53 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*}

✓ Solution by Maple

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

dsolve([diff(x(t),t) = -y(t), diff(y(t),t) = 13*x(t)+4*y(t), x(0) = 0, y(0) = 3], singsol=all)
 

\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*}

✓ Solution by Mathematica

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

DSolve[{D[x[t],t]==-y[t],D[y[t],t]==13*x[t]+4*y[t]},{x[0]==0,y[0]==3},{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*}