76.8.13 problem 13

Internal problem ID [17502]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 3. Systems of two first order equations. Section 3.4 (Complex Eigenvalues). Problems at page 177
Problem number : 13
Date solved : Tuesday, January 28, 2025 at 10:39:37 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.072 (sec). Leaf size: 37

dsolve([diff(x(t),t)=a*x(t)+y(t),diff(y(t),t)=-x(t)+a*y(t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{a t} \left (c_{1} \sin \left (t \right )+c_{2} \cos \left (t \right )\right ) \\ y \left (t \right ) &= -{\mathrm e}^{a t} \left (c_{2} \sin \left (t \right )-\cos \left (t \right ) c_{1} \right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 43

DSolve[{D[x[t],t]==a*x[t]+y[t],D[y[t],t]==-x[t]+a*y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to e^{a t} (c_1 \cos (t)+c_2 \sin (t)) \\ y(t)\to e^{a t} (c_2 \cos (t)-c_1 \sin (t)) \\ \end{align*}