satisfy a solution of an equation

[Dolphin2022]

Hi,
I have been using the diffusion equation based on Ficks 1st, 2nd law and continuity. It has the form

dy/dt = D*(d^2y/dx^2) (equation 1)

where d = partial derivative, D= diffusion coefficient, y= flux. It has the solution:

y(x,t)=1/sqrt(4piDt)*exp^[-x^2/4Dt] (equation 2)

If you plot this it is a decaying curve such that y-->0 as x-->infinity. Its not too difficult to show that equation 2 satisfies equation 1.

The thing is, i'm using a growth version of this equation, which has a positive in the exponential of the form

y(x,t)=1/sqrt(4piDt)*exp^[+x^2/4Dt] (equation 3)

If you plot this it is a growth curve such that y-->infinity as x-->infinity. But when I try to go through the proof and show that equation 3 also satisfies equation 1 its not quite the same. It may just be I have a minus missing somewhere. If I could get any help on this it would be really great. I just want to know if equation 3 satisfies equation 1?

thanks

 

[MathLover1]

this is what I found for you and think it might be helpful:

Note that the integral of any function f(x) between x and x +dx is simply f(x) dx!