Internal problem ID [9437]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 8, system of first order odes
Problem number: 1858.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\relax (t )&=a y \relax (t )\\ y^{\prime }\relax (t )&=b x \relax (t ) \end {align*}
✓ Solution by Maple
Time used: 0.077 (sec). Leaf size: 64
dsolve({diff(x(t),t)=a*y(t),diff(y(t),t)=b*x(t)},{x(t), y(t)}, singsol=all)
\[ x \relax (t ) = c_{1} {\mathrm e}^{\sqrt {a}\, \sqrt {b}\, t}+c_{2} {\mathrm e}^{-\sqrt {a}\, \sqrt {b}\, t} \] \[ y \relax (t ) = \frac {\sqrt {b}\, \left (c_{1} {\mathrm e}^{\sqrt {a}\, \sqrt {b}\, t}-c_{2} {\mathrm e}^{-\sqrt {a}\, \sqrt {b}\, t}\right )}{\sqrt {a}} \]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 94
DSolve[{x'[t]==a*y[t],y'[t]==b*x[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to c_1 \cosh \left (\sqrt {a} \sqrt {b} t\right )+\frac {\sqrt {a} c_2 \sinh \left (\sqrt {a} \sqrt {b} t\right )}{\sqrt {b}} \\ y(t)\to c_2 \cosh \left (\sqrt {a} \sqrt {b} t\right )+\frac {\sqrt {b} c_1 \sinh \left (\sqrt {a} \sqrt {b} t\right )}{\sqrt {a}} \\ \end{align*}