# PDE with substitution - proving the correctness of the substitution

I have the differential equation:

$$\frac{\partial C}{\partial t} = D \frac{\partial^2 C}{\partial z^2}$$

to solve it, the author of my book carries out the following replacement:

$$-2\zeta \frac{dX}{d \zeta} = \frac{d^2 X}{d \zeta^2}$$

with $X=1-\frac{C}{C_S}$ and $\zeta =\frac{z}{(4Dt)^{1/2}}$.

How can I prove that the last equation is identical to the first in Mathematica?

Thank you so much for your time.

• Could you please describe the substitution z->zeta in detail? What means Dt? – Ulrich Neumann Jan 5 '18 at 11:12
• Hello @UlrichNeumann, D is a constant, z and t are variables. – Gennaro Arguzzi Jan 5 '18 at 11:35

pde = Derivative[1, 0][C][t, z] == d Derivative[0, 2][C][t, z]