73.27.12 problem 38.10 (f)

Internal problem ID [15768]
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)
Date solved : Tuesday, January 28, 2025 at 08:06:52 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=3 x \left (t \right )+2 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-2 x \left (t \right )+3 y \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x \left (0\right ) = a_{1}\\ y \left (0\right ) = a_{2} \end{align*}

✓ Solution by Maple

Time used: 0.043 (sec). Leaf size: 44

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 (-\sin \left (2 t \right ) a_{1} +\cos \left (2 t \right ) a_{2} \right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 47

DSolve[{D[x[t],t]==3*x[t]+2*y[t],D[y[t],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*}