Are those distributional solutions that are functions, the same as weak solutions?

There are two closely related concepts and I am not sure exactly how close. Consider a partial differential equation. (The coefficients need not be constant but assume they are functions, and not more general distributions):

A weak solution is a function that satisfies the weak form of the equation.

A distributional solution is a distribution that satisfies the equation itself, when derivatives are understood as distributional derivatives. (As Alexandre Eremenko remarks in a comment, this will not always make sense since you cannot always multiply distributions. But it does make sense in the case that the distribution is a function, which is the case I ask about.)

The second concept is actually broader since, for example the Dirac delta $delta$ is a distributional solution of the ordinary differential equation (in variable $x$) $xf’=-f$ but not a weak solution since it is not a function.

What I do not know is this: When a function is a distributional solution to an equation then is it always a weak solution? And vice versa?

I believe I can show the answer is yes in both cases for linear partial differential equations, by fairly direct calculation. But I fear I may be missing some nuance. And I am not at all sure how to proceed for non-linear PDEs.