1. ## heat equation

1. Show that the heat equation has complex-valued solutions of the form
F(x) exp(iwt) provided that kF′′ = iwF.

Done This

2. Find F if F(x) -> 0 as
x -> infinity and F(0) = A, where A is a real constant. Let T(x, t) be the
real part of F(x) exp(iwt). Verify that

T(x, t) = Aexp(−((w/2k)^1/2)x)cos(wt − ((w/2k)^1/2)x)

and that T is a solution of the heat equation for which T(x, t) --> 0 as
x --> inifinity and T(0, t) = Acoswt. Can you think of a physical situation
to which T corresponds?

I am not sure how to start this second bit, any hints would be much appreciated

many thanks

2. The equation $\displaystyle kF'' = i\omega F$ is a standard SHM equation, with solutions $\displaystyle F(x) = Ae^{\gamma x} + Be^{-\gamma x}$, where $\displaystyle \gamma^2 = i\omega/k$. Notice that to find $\displaystyle \gamma$ you need to find the square root of $\displaystyle i\omega/k$. For that, remember that $\displaystyle i = e^{i\pi/2}$, which has a square root $\displaystyle e^{i\pi/4}$.