Internal problem ID [14032]
Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell.
second edition. CRC Press. FL, USA. 2020
Section: Chapter 38. Systems of differential equations. A starting point. Additional Exercises.
page 786
Problem number: 38.10 (f).
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\left (t \right )&=3 x \left (t \right )+2 y \left (t \right )\\ y^{\prime }\left (t \right )&=-2 x \left (t \right )+3 y \left (t \right ) \end {align*}
With initial conditions \[ [x \left (0\right ) = a_{1}, y \left (0\right ) = a_{2}] \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 45
dsolve([diff(x(t),t) = 3*x(t)+2*y(t), diff(y(t),t) = -2*x(t)+3*y(t), x(0) = a__1, y(0) = a__2], singsol=all)
\begin{align*} x \left (t \right ) &= {\mathrm e}^{3 t} \left (a_{2} \sin \left (2 t \right )+a_{1} \cos \left (2 t \right )\right ) \\ y \left (t \right ) &= {\mathrm e}^{3 t} \left (a_{2} \cos \left (2 t \right )-a_{1} \sin \left (2 t \right )\right ) \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 47
DSolve[{x'[t]==3*x[t]+2*y[t],y'[t]==-2*x[t]+3*y[t]},{x[0]==a1,y[0]==a2},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to e^{3 t} (\text {a1} \cos (2 t)+\text {a2} \sin (2 t)) \\ y(t)\to e^{3 t} (\text {a2} \cos (2 t)-\text {a1} \sin (2 t)) \\ \end{align*}