Existence and Uniqueness theorem for 1st order ODEs

[Silversonic]

Homework Statement

Consider the IVP compromising the ODE.

dy/dx = sin(y)

subject to the initial condition y(X) = Y

Without solving the problem, decide if this initial value problem is guaranteed to have a unique solution. If it does, determine whether the existence of that solution is guaranteed for all values of x.

I'm not sure how to answer this. f(x,y) and df(x,y)/dy are both continuous for all values of x and y. This means there is exactly one solution to the IVP.

Now working out what the solution is, we get;

In(1-cos(y)/sin(y)) = x + C.

What I don't get is whether it's guaranteed for all values of x? I don't believe it is, as because if x < -C then we get a negative number on the RHS. This is not computable. HOWEVER, if I'm right, how was I meant to work that out without working out the solution?

I also need to check that I'm correct in saying that that existence of a unique solution is guaranteed.

 

[Silversonic]

b-b-b-b-ump