Q. Determine whether the initial value problem [tex]y' = x^5 - y^5 + 2xe^y ,y\left( 3 \right) = \pi [/tex] has a unique solution. 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.