I would like to know if my solution is correct.
for problem 1. 3
I am supposed to solve $$u_{t} = ku_{xx}, u(x, 0) = g(x)$$
where $$g(x) = \exp(-a |x|)$$
My solution:
$$u(x, t) = \frac{1}{4\sqrt{kt\pi}} \int_{-\inf}^{\inf} \exp(\frac{-(x-y)^2}{4kt}) \exp(-a|y|) dy$$
(*) to get rid of the absolute value:
$$u(x, t) = \frac{2}{4\sqrt{kt\pi}} \int_{0}^{\inf} \exp(\frac{-(x-y)^2}{4kt}) \exp(-ay) dy$$
I then complete the square for: $\frac{-(x-y)^2 - 4ktay}{4kt}$
to get:
$$u(x, t) = 2 \exp(ax-a^2kt) \frac{1}{4\sqrt{kt\pi}} \int_{0}^{\inf} \exp(\frac{-(y+2kat-x)^2}{4kt}) dy$$
I then use: $$1 = \frac{1}{4\sqrt{kt\pi}} \int_{-\inf}^{\inf} \exp(\frac{-(y+2kat-x)^2}{4kt}) dy$$
(**) to write: $$\frac{1}{2} = \frac{1}{4\sqrt{kt\pi}} \int_{0}^{\inf} \exp(\frac{-(y+2kat-x)^2}{4kt}) dy$$
$$u(x, t) = \frac{2 \exp(ax-a^2kt)}{2} = \exp(ax-a^2kt) $$
I feel like I might have taken a wrong turn at step (*) and/or (**); could someone let me know if this is right or wrong
