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 : Wednesday, February 05, 2025 at 03:45:56 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*}

✓ Solution by Maple

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

dsolve([diff(x(t),t) = x(t)+9*y(t), diff(y(t),t) = -2*x(t)-5*y(t), x(0) = 3, y(0) = 2], singsol=all)
 

\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*}

✓ Solution by Mathematica

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

DSolve[{D[x[t],t]==x[t]+9*y[t],D[y[t],t]==-2*x[t]-5*y[t]},{x[0]==3,y[0]==2},{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*}