Internal problem ID [9440]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 8, system of first order odes
Problem number: 1861.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\relax (t )&=\frac {a x \relax (t ) \alpha +a y \relax (t ) \beta +x \relax (t ) b \beta -\alpha y \relax (t ) b}{a^{2}+b^{2}}\\ y^{\prime }\relax (t )&=-\frac {a x \relax (t ) \beta -a \alpha y \relax (t )-x \relax (t ) \alpha b -y \relax (t ) b \beta }{a^{2}+b^{2}} \end {align*}
✓ Solution by Maple
Time used: 0.106 (sec). Leaf size: 144
dsolve({a*diff(x(t),t)+b*diff(y(t),t)=alpha*x(t)+beta*y(t),b*diff(x(t),t)-a*diff(y(t),t)=beta*x(t)-alpha*y(t)},{x(t), y(t)}, singsol=all)
\[ x \relax (t ) = c_{1} {\mathrm e}^{\frac {\left (i a \beta -i \alpha b +a \alpha +b \beta \right ) t}{a^{2}+b^{2}}}+c_{2} {\mathrm e}^{-\frac {\left (i a \beta -i \alpha b -a \alpha -b \beta \right ) t}{a^{2}+b^{2}}} \] \[ y \relax (t ) = i \left (c_{1} {\mathrm e}^{\frac {\left (i a \beta -i \alpha b +a \alpha +b \beta \right ) t}{a^{2}+b^{2}}}-c_{2} {\mathrm e}^{-\frac {\left (i a \beta -i \alpha b -a \alpha -b \beta \right ) t}{a^{2}+b^{2}}}\right ) \]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 144
DSolve[{a*x'[t]+b*y'[t]==\[Alpha]*x[t]+\[Beta]*y[t],b*x'[t]-a*y'[t]==\[Beta]*x[t]-\[Alpha]*y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to e^{\frac {t (a \alpha +b \beta )}{a^2+b^2}} \left (c_1 \cos \left (\frac {t (a \beta -\alpha b)}{a^2+b^2}\right )+c_2 \sin \left (\frac {t (a \beta -\alpha b)}{a^2+b^2}\right )\right ) \\ y(t)\to e^{\frac {t (a \alpha +b \beta )}{a^2+b^2}} \left (c_2 \cos \left (\frac {t (a \beta -\alpha b)}{a^2+b^2}\right )+c_1 \sin \left (\frac {t (\alpha b-a \beta )}{a^2+b^2}\right )\right ) \\ \end{align*}