60.9.31 problem 1886

Internal problem ID [11810]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 8, system of first order odes
Problem number : 1886
Date solved : Sunday, March 30, 2025 at 09:15:50 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*}

✓ Maple. Time used: 0.140 (sec). Leaf size: 49

ode:=[diff(diff(x(t),t),t)+a*y(t) = 0, diff(diff(y(t),t),t)-a^2*y(t) = 0]; 
dsolve(ode);
 

\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*}

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 103

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

✓ Sympy. Time used: 0.229 (sec). Leaf size: 46

from sympy import * 
t = symbols("t") 
a = symbols("a") 
x = Function("x") 
y = Function("y") 
ode=[Eq(a*y(t) + Derivative(x(t), (t, 2)),0),Eq(-a**2*y(t) + Derivative(y(t), (t, 2)),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = C_{1} + C_{2} t + \frac {C_{3} e^{- a t}}{a^{2}} - \frac {C_{4} e^{a t}}{a^{2}}, \ y{\left (t \right )} = - \frac {C_{3} e^{- a t}}{a} + \frac {C_{4} e^{a t}}{a}\right ] \]