In part of dealing with solving a differential equation, ie., $$\frac{\partial u}{\partial t }-\frac{\partial^2 u}{\partial x^2}=f(x,t)$$
(Obviously with given Initial/Boundary conditions with the PDE) We can expand the function $u(x,t)$ as a fourier series as follows.
$$ u(x,t)= \Sigma^\infty_{n=0} a_n(t) \sin(\frac{\pi n}{\ell} x);\ a_n(t)=\frac 2\ell \int^t_{t_0} u(x,t) sin(\frac{\pi n }{\ell}x )\ \mathrm{d} x $$ And, similarly, expand $f(x,t)$ the same way
$$ f(x,t)= \Sigma^\infty_{n=0} b_n(t) \sin(\frac{\pi n}{\ell} x);\ b_n(t)=\frac 2\ell \int^t_{t_0} f(x,t) sin(\frac{\pi n }{\ell} x)\ \mathrm{d} x $$
The book I am reading says that you can determine $b_n(t)$ from this expansion of $f$, because we're going to end up solving a system of differential equations of the form $\frac{\mathrm{d}a_n (t)}{\mathrm{d}t}-\alpha\ a_n(t)=\beta\ b_n(t);\ a_n(t_0)=c_n$. Is this just because in $\int_a^b f(x,t)\ \mathrm{d}t$, the $t$ variable will still remain? I think this is correct, but wanted to double check with someone before just assuming that's how it's done.
I don't think I've ever had to handle integrating a multivariate function with respect to one variable, without dealing with the second variable as well (ie., double integrals, given an area such as $A:\{ 0,1\} ^2 \subset \mathbb{R}^2$, calculate $\int_A f(x,y)\ \mathrm{d}A=\int_0^1\int_0^1 f(x,y)\ \mathrm{d}x\mathrm{d}y $). Is it really just that simple? For example, if we were calcuting $\int_0^{2\pi} xt\exp(x^3t^{-2})\ \mathrm{d}x$, we just act as if $t$ is a constant and deal with the result after the integration (Pretty much leaving us with $t\exp(\frac1{t^2})\int_0^{2\pi} x\exp(x^3) dx$ if I remember my integration properties correctly)?
It's been awhile since I've formally did any maths, so sorry if the answer to this question is yes ( which I think it is), but you guys are much more versed in this area than I ever could be, and I'd rather double check and look silly than be incorrect in thinking this is the process and it not really being so.
Thanks in advance!!