Chapter 5.3, Problem 80E

The Error Function The error function $\mathrm{erf}(x)=\frac{2}{\sqrt{\pi }}{\displaystyle {\int }_0^x{e}^{-t^2}}dt$ is used in probability, statistics, and engineering.

(a) Show that ${\displaystyle {\int }_a^b{e}^{-t^2}}dt=\frac{1}{2}\sqrt{\pi }[\mathrm{erf}(b)-\mathrm{erf}(a)]$ .

(b) Show that the function $y=e^{x^2}\mathrm{erf}(x)$ satisfies the differential equation $y\prime =2xy+2/\sqrt{\pi }$ .