So I am doing a course this semester in PDEs and we are currently doing the heat/diffusion equation $(u_t +ku_{xx}=0)$ on the whole line and the half line. In solving these equations we have introduced the error function,$$\operatorname{erf}(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2}dt.$$
My question is
What exactly is the error function and why is it natural or useful to express a solution to a PDE in terms of it?
I understand that it is closely related to the normal cdf $$ P(N < x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^x e^{-t^2/2}dt$$ but it doesn't seem to make a lot of sense when solving PDEs. I'm just questioning why, when I have a perfectly good solution, do I have to then go and make substitutions and change my limits of integration just to put it in terms of this error function. I have to assume that there is some logic behind this and that it was not invented just to annoy students in the future, like myself.
Thank you for any help!