- #1
Silversonic
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- 1
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.
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