60.9.4 problem 1859

Internal problem ID [11858]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 8, system of first order odes
Problem number : 1859
Date solved : Monday, January 27, 2025 at 11:43:52 PM
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: 0.066 (sec). Leaf size: 36

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_{2} \cos \left (t \right )+c_{1} \sin \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*}