Differential equation solved with variables separable method
Explain the steps on how to solve the problem below:
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.