Given the differential equation $$ \dfrac{dy}{dx} = 2y + |x|, \forall (x, y) \in [-4, 4] \times [-4, 4]. $$ Explain, with calculations, why the differential equation has no solutions in Hilbert space but rather in the Sobolev spaces.
I have done the calculations, but I don't know how to relate it to Hilbert and Sobolev spaces. I have no idea about how to do differential equations in Hilbert and Sobolev spaces. Can someone help show me how to answer this question? Also, can anyone suggest a good reference for solving differential equations in Hilbert and Sobolev spaces?