Weak Solution to Dx u = u

Theorem
Let $H$ be the Heaviside step function.

Let $\map u {x, t} = \map H t e^x$

Then $u$ is a weak solution of the partial differential equation $\ds \dfrac {\partial u} {\partial x} = u$.

That is, for the distribution $T_u \in \map {\DD'} {\R^2}$ associated with $u$ in the distributional sense it holds that:


 * $\ds \dfrac {\partial T_u}{\partial x} = T_u$

Proof
Let $\phi \in \map \DD {\R^2}$ be a test function.

Then in the distributional sense we have that: