Solution to the Heat Equation?

If u(x,t) is the temperature in a rod at time t a distance x from some fixed point, then to a good approximation u(x,t) satisfies the Heat Equation:

du/dt = D(d2u/dx2) where D is a constant which describes physical properties of the rod. For t > 0, define u(x,t) = e^(-kx^2/t)/sqrt(t) where k is a constant. There is one value of k for which this function is a solution of the Heat Equation. Find the value of k and verify that the resulting function does solve the equation. The value of k will be related to D.

Please help!! I know it's a famous solution but I don't understand it.

Since u(x, t) = t^(-1/2) e^(-kx^2/t), we have

u_t = (-1/2)t^(-3/2) * e^(-kx^2/t) + t^(-1/2) *(kx^2/t^2) e^(-kx^2/t)
......= [(-1/2)t^(-3/2) + kx^2 t^(-5/2)] e^(-kx^2/t).

u_x = t^(-1/2) * (-2kx/t) e^(-kx^2/t)
.......= -2kxt^(-3/2) e^(-kx^2/t)

u_xx = -2kt^(-3/2) e^(-kx^2/t) + -2kxt^(-3/2) * (-2kx/t) e^(-kx^2/t)
........= [-2kt^(-3/2) + 4k^2 x^2 t^(-5/2)] e^(-kx^2/t)

So, u_t = D u_xx
==> [(-1/2)t^(-3/2) + kx^2 t^(-5/2)] e^(-kx^2/t)
= D [-2kt^(-3/2) + 4k^2 x^2 t^(-5/2)] e^(-kx^2/t)

Simplifying yields
(-1/2)t^(-3/2) + kx^2 t^(-5/2) = -2kDt^(-3/2) + 4k^2D x^2 t^(-5/2)

So, -1/2 = -2kD and k = 4k^2D
==> k = 1/(4D).

I hope this helps!