Can anyone help with the following maths question:
Show that the function:
u(t,x)=f(y), where y=((2x)/sqrt(t))
is a solution of the heat equation:
partialderivative(u) / partialderivative(t) = partialderivative(u)^2 / partialderivative(x)^2
(where the partialderivative(u or x)^2 is the second derivative)
provided that f(y) satisfies 8f''(y) + yf'(y)=0
Thanks in advance!
All my ds in the following are partials.
I got that du/dt = -f'(y)xt^(-3/2)
I then found that du/dx = 2f'(y)t^(-1/2)
From this I said that d^2u/dx^2 = 2f''(y)t^(-1/2)
I then re-arranged 8f''(y) + yf'(y) = 0 to f''(y) = (-yf'(y))/8
Substituting into d^2u/dx^2 gives (-yf'(y)t^(-1/2))/4
Letting y = 2xt^-(1/2) means d^2u/dx^2 = (-f'(y)xt^(-1))/2
Guessing I'm doing something stupid somewhere?!