60.9.31 problem 1886

Internal problem ID [11885]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 8, system of first order odes
Problem number : 1886
Date solved : Tuesday, January 28, 2025 at 06:24:00 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d^{2}}{d t^{2}}x \left (t \right )+a y \left (t \right )&=0\\ \frac {d^{2}}{d t^{2}}y \left (t \right )-a^{2} y \left (t \right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.103 (sec). Leaf size: 49

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: 0.009 (sec). Leaf size: 103

DSolve[{D[x[t],{t,2}]+a*y[t]==0,D[y[t],{t,2}]-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*}