1. Mathematics
2. Numerical Analysis
3. Ordinary Differential Equations
4. 59710

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Let x: [0, infinity) -> R and y: [0, infinity) -> R be solutions to the system of differential equations:

x' = - x

y' = - sin y

With initial condition:

x(0) = y(0) = alpha, where alpha belongs to [0, pi)

(a) Show that |x(t)| =< alpha for all t >= 0

(b) Show that | y(t) - x(t) | =< alpha( 1 - e^-t) for all t >= 0. ( e here is exponential function)

Please justify every step and claim, and if you use any theorems refer to them.