Periodic solution of PDE

Question

I am little confused in how to solve this equation :

$$u +u^{''}=0$$ where $u$ is real $2\pi-$periodic function and the derivative is in the sens of distributions

So my question is that what is the solution of this PDE

or what is the space of the solution of this PDE

If we have more regularity we can solve this with fourier series but in my case where I have this I dont know how to solve it thank you in advance

Answer 1

Both $u_1 = sin(x)$ and $u_2 = cos(x)$ solve this equation.

Because: $u''_1= -sin(x)$ and $u''_2= -cos(x)$, thus we have either: $$u_1 + u''_1 = 0$$ $$sin(x) + -sin(x) = 0$$ Or: $$u_2 + u''_2 = 0$$ $$cos(x) + -cos(x) = 0$$

I hope it helped :)