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I've seen things like Lipschitz bound and conditions on the partial derivative of f(x,y) with respect to y where y' = f(x,y) but such things mean very little to me. The reason is because I've never seen any examples where something like a Lipschitz bound is applied. I've also yet to see the df/dy(partial) continuity conditions applied. For this example if I take the RHS to be f(x,y) then obviously the partial derivative of f wrty is continuous everywhere so the additional conditions which come with df/dy don't really apply.

I'm just wondering if the IVP has a unique solution. My limited knowledge of DEs is inclined to suggest to me that there is a unique solution to the DE because the partial derivative of the RHS with respect to y is continuous everywhere. Can someone offer some input? Any help appreciated.