Explicit weak solution of a first-order evolution equation with periodic condition

Question

I am reading the topic on weak solutions of first-order ODEs, and I am wondering whether there is a closed-form formulas for the solution in some concrete scenarios. For example,

given $f\in L^2(\left[0,T\right])$, it is know that the equation $$u'(t)+u(t)=f(t)\;\text{a.e. on}\;\left]0,T\right[\quad\text{and}\quad u(0)=u(T) $$ admits a unique solution in $C(\left[0,T\right])\cap W^{1,2}(\left[0,T\right])$.

My question: Can we explicitly solve for $u$ in terms of $f$? It seems that the techniques used when the condition is $u(0)=u_0$ does not apply here.

Any help/hint is highly appreciated.

Answer 1

You apply exactly the same techniques. Apply the integrating factor and integrate $$ e^tu(t)-u(0)=\int_0^te^sf(s)\,ds $$ The right side is a continuous function, and it should be easy to check also in $W^{1,2}$. To get the periodicity condition the initial value has to satisfy $$ (e^T-1)u(0)=\int_0^Te^sf(s)\,ds $$ which has a unique solution.