9.4 problem 1859

Internal problem ID [10191]

Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 8, system of first order odes
Problem number: 1859.
ODE order: 1.
ODE degree: 1.

Solve \begin {align*} x^{\prime }\left (t \right )&=a x \left (t \right )-y \left (t \right )\\ y^{\prime }\left (t \right )&=x \left (t \right )+a y \left (t \right ) \end {align*}

Solution by Maple

Time used: 0.156 (sec). Leaf size: 37

dsolve({diff(x(t),t)=a*x(t)-y(t),diff(y(t),t)=x(t)+a*y(t)},{x(t), y(t)}, singsol=all)
 

\[ x \left (t \right ) = {\mathrm e}^{a t} \left (c_{1} \sin \left (t \right )+c_{2} \cos \left (t \right )\right ) \] \[ y \left (t \right ) = {\mathrm e}^{a t} \left (\sin \left (t \right ) c_{2} -\cos \left (t \right ) c_{1} \right ) \]

Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 43

DSolve[{x'[t]==a*x[t]-y[t],y'[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*}