Trouble with dirac delta in R^2

[obomov2]

Find a distribution F in R^2 that satisfies (Dx) F(x,t) = t*Delta(x)
It is apperantly not t*H(x) as in R.

* is multiplication, D is dirac delta, H is Heavyside , (Dx) is derivation with respect to x (in the sense of distributions)

Sorry for not using Latex.

Indeed I am trying to solve following ODE for u(t,x) :
u' + u = t*Delta(x) + Delta'' (x)

 

[gel]

Find a distribution F in R^2 that satisfies (Dx) F(x,t) = t*Delta(x)
It is apperantly not t*H(x) as in R.

why not?

Indeed I am trying to solve following ODE for u(t,x) :
u' + u = t*Delta(x) + Delta'' (x)

multiply both sides by exp(x) to get d/dx(exp(x)u) on the left hand side and integrate.