Find the Fourier series solution to the differential equation

[Paradoxx]

Find the Fourier series solution to the differential equation x"+x=t

It's given that x(0)=x(1)=0

So, I'm trying to find a Fourier serie to x(t) and f(t)=t, and I'm know it must a serie of sin...



So here's my question...the limits of integration to the Bn, how do I define them? Will it be like 0 to L to both series? And about the x", after a I find the f(x) Fourier series I must just derive it and replace in the x"??

 

[Paradoxx]

t = Ʃ Tn sin (n∏x/L)

where f(t) = t = 2/L ∫ f(t) sin(n∏x/L)

If the period 2L = 2, my limits on the integral will be 0 to 1?

 

[vanhees71]

Since the right-hand side of the equation is not periodic, you have to use a Fourier integral rather than a Fourier series. Further one has to regularize the right-hand side, because it's not a Fourier-transformable function. I guess that the idea is that the external force is switched on at $t=0$. So I'd write
$$t \rightarrow \Theta(t) t \exp(-\epsilon t).$$
Then you can evaluate the Fourier transform of both the left-hand side and the right-hand side of the equation. At the end of the calculation, after transforming back to the time domain, you can take $\epsilon \rightarrow 0^+$.

I also don't understand, why you have given boundary conditions at t=0 and t=1 rather than an initial condition $x(t=0)=x_0$, $\dot{x}(t=0)=v_0$. This you could solve by first finding a particular solution of the inhomogeneous equation (using the Fourier-integral ansatz) and then add the general solution of the homogeneous equation.