Diffusion Equations and Boundary Conditions

Hello, I have a test coming up soon and I don't understand how to solve the kind of problem as given below. Please provide some explanation in the solution. Thanks in advance,

Eriko

Consider 
the
 diffusion
 equation

φt = a
2
φxx + x
3
, −L ≤ x ≤ L
with
 initial 
condition


φ(x,0) =1
(i) Suppose 
the 
boundary 
conditions 
are


φ(−L,t) = 0 φ(L,t) = 0
Does 
an 
equilibrium 
solution 
exist?
 If so,
solve
 for
 it.
(ii) Suppose
 the 
boundary 
conditions 
are

φ(−L,t) = 0 φx (L,t) = 0
Does
 an
 equilibrium 
solution
 exist?
 If 
so,
 solve 
for 
it.
(iii) Suppose 
the 
boundary 
conditions
 are

φ(−L,t) =1 φx (L,t) = 0
Does
 an
 equilibrium 
solution exist? 
If
 so, 
solve 
for 
it.

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Solution Preview

...is a function that only depends on the coordinate x.
Thus, our equation
(1.2)
Becomes a simple second order ordinary differential equation:
(1.3)
The homogenous equation is:
(1.4)
Which leads to the simple homogenous solution:
(1.5)
For the particular solution, we see that the second derivative should have a term of , so we "guess" a solution in the form:
(1.6)
Substituting ...