60.9.6 problem 1861

Internal problem ID [11860]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 8, system of first order odes
Problem number : 1861
Date solved : Monday, January 27, 2025 at 11:43:54 PM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.143 (sec). Leaf size: 134

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)},singsol=all)
 

\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{\frac {\left (i a \beta -i b \alpha +a \alpha +b \beta \right ) t}{a^{2}+b^{2}}}+c_{2} {\mathrm e}^{-\frac {\left (i a \beta -i b \alpha -a \alpha -b \beta \right ) t}{a^{2}+b^{2}}} \\ y \left (t \right ) &= i \left (c_{1} {\mathrm e}^{\frac {\left (i a \beta -i b \alpha +a \alpha +b \beta \right ) t}{a^{2}+b^{2}}}-c_{2} {\mathrm e}^{-\frac {\left (i a \beta -i b \alpha -a \alpha -b \beta \right ) t}{a^{2}+b^{2}}}\right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 145

DSolve[{a*D[x[t],t]+b*D[y[t],t]==\[Alpha]*x[t]+\[Beta]*y[t],b*D[x[t],t]-a*D[y[t],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 (a \beta -\alpha b)}{a^2+b^2}\right )\right ) \\ \end{align*}