can be solved by the displacement equation , or , or even . To show this, enter the displacement equation and then show that both sides of the differential equation are equal.

>f:=(x,t)->sin(x+a*t); >g:=(x,t)->sin(k*x)*cos(k*a*t); >h:=(x,t)->sin(x+a*t)+37.8; >diff(f(x,t),t,t)-a^2*diff(f(x,t),x,x); >diff(g(x,t),t,t)=a^2*diff(g(x,t),x,x); >diff(h(x,t),t,t)=a^2*diff(h(x,t),x,x);

- Fick's second law of diffusion satisfies the partial differential equation

The function

denotes diffusion of a substance at a depth and time where is the initial amount of the substance and is the diffusion coefficient.**A)**- Enter the function and show that the function satisfies the partial differential equation given above. (Note: you may need to use the
**simplify**command on both sides of the equal sign.) **B)**- If the initial amount of is
and the diffusion coefficients of is in the air and in the water, plot the surface concentration (N(0,t)) for ten seconds. There should be two plots; one titled
*diffusion through air*and the other titled*diffussion through water*. **C)**- Does the surface concentration increase or decrease for each as time increases? Explain why this is so. Which concentration is higher over time? Explain why this is so.

- Determine which of the following functions satisfy Laplace's equation

**A)****B)****C)****D)**

- Using a function from exercise 2 that you found satisfies the Laplace equation, answer the following without calculating the differential equation.
**A)**- Will the function plus satisfy the Laplace equation? Why or why not?
**B)**- Will the function plus satisfy the Laplace equation? Why or why not?
**C)**- Will the function plus satisfy the Laplace equation? Why or why not?

2006-11-01