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.

  • $\begingroup$ 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. $\endgroup$ – PradyumanDixit Oct 11 '18 at 12:42
  • $\begingroup$ Hello, done as required. $\endgroup$ – ssheep Oct 11 '18 at 13:55
  • $\begingroup$ Why do you want a distributional solution? You can easily compute a piecewise differentiable solution from the general solutions of $u'+tu=0$ and $u'+tu=1$. $\endgroup$ – LutzL Oct 11 '18 at 14:08
  • $\begingroup$ 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. $\endgroup$ – ssheep Oct 11 '18 at 14:13
  • $\begingroup$ 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. $\endgroup$ – Ian Oct 11 '18 at 14:19

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