60.9.3 problem 1858

Internal problem ID [11857]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 8, system of first order odes
Problem number : 1858
Date solved : Monday, January 27, 2025 at 11:43:51 PM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.063 (sec). Leaf size: 63

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

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

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 158

DSolve[{D[x[t],t]==a*y[t],D[y[t],t]==b*x[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {e^{-\sqrt {a} \sqrt {b} t} \left (\sqrt {b} c_1 \left (e^{2 \sqrt {a} \sqrt {b} t}+1\right )+\sqrt {a} c_2 \left (e^{2 \sqrt {a} \sqrt {b} t}-1\right )\right )}{2 \sqrt {b}} \\ y(t)\to \frac {e^{-\sqrt {a} \sqrt {b} t} \left (\sqrt {b} c_1 \left (e^{2 \sqrt {a} \sqrt {b} t}-1\right )+\sqrt {a} c_2 \left (e^{2 \sqrt {a} \sqrt {b} t}+1\right )\right )}{2 \sqrt {a}} \\ \end{align*}