# Distributional solution.

How to solve the following differential equation on the space of distributions? $$u'+tu=K(0,1),$$ Here $$K_{(0,1)}$$ is the characteristic function of the closed interval between $$0$$ and $$1$$. I managed to find out that the problem is to find a $$u$$ such that the following integral equation is satisfied:

$$\int u(t) (-exp(t^2/2)) (exp(-t^2/2) p(t) )'dt=\int K_{(0,1)}(t) p(t)dt$$

where $$p$$ ranges over infinitely differential functions of compact support, and integration is on the line.

• Hello, welcome to MSE, Kindly refrain yourself from posting questions without any effort of your own, Kindly attach your own method or your trial to improve the question. – PradyumanDixit Oct 11 '18 at 12:42
• Hello, done as required. – ssheep Oct 11 '18 at 13:55
• Why do you want a distributional solution? You can easily compute a piecewise differentiable solution from the general solutions of $u'+tu=0$ and $u'+tu=1$. – LutzL Oct 11 '18 at 14:08
• That is true but this is an exercise. I should find the classical solutions of the homogenic problem, then find one particular solution on the space of distributions. These latter objects additively form the general solution of the problem. – ssheep Oct 11 '18 at 14:13
• The easiest way to do that is to just write down the piecewise differentiable solution as @LutzL suggested and then check that it is a weak solution of the full problem. – Ian Oct 11 '18 at 14:19