72.9.13 problem 28

Internal problem ID [14809]
Book : DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th edition. Brooks/Cole. Boston, USA. 2012
Section : Chapter 3. Linear Systems. Exercises section 3.1. page 258
Problem number : 28
Date solved : Tuesday, January 28, 2025 at 07:17:03 AM
CAS classification : system_of_ODEs

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

With initial conditions

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

✓ Solution by Maple

Time used: 0.054 (sec). Leaf size: 43

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

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-2 t} \left (-3 \sin \left (3 t \right )+2 \cos \left (3 t \right )\right ) \\ y &= {\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: 0.003 (sec). Leaf size: 46

DSolve[{D[x[t],t]==-2*x[t]-3*y[t],D[y[t],t]==3*x[t]-2*y[t]},{x[0]==2,y[0]==3},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

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