Internal problem ID [10209]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 8, system of first order odes
Problem number: 1886.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime \prime }\left (t \right )+a y \left (t \right )&=0\\ y^{\prime \prime }\left (t \right )-a^{2} y \left (t \right )&=0 \end {align*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 50
dsolve([diff(x(t),t,t)+a*y(t)=0,diff(y(t),t,t)-a^2*y(t)=0],singsol=all)
\begin{align*} x \left (t \right ) &= -\frac {-c_{1} t a +c_{3} {\mathrm e}^{-a t}+c_{4} {\mathrm e}^{a t}-c_{2} a}{a} \\ y \left (t \right ) &= c_{3} {\mathrm e}^{-a t}+c_{4} {\mathrm e}^{a t} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.016 (sec). Leaf size: 103
DSolve[{x''[t]+a*y[t]==0,y''[t]-a^2*y[t]==0},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {c_4 \left (2 a t+e^{-a t}-e^{a t}\right )}{2 a^2}-\frac {c_3 e^{-a t} \left (e^{a t}-1\right )^2}{2 a}+c_2 t+c_1 \\ y(t)\to \frac {e^{-a t} \left (a c_3 \left (e^{2 a t}+1\right )+c_4 \left (e^{2 a t}-1\right )\right )}{2 a} \\ \end{align*}