Confused about separation of variables for PDE

[jetforcegemin]

So my book says that to solve a PDE by separation of variables, we check the three cases where λ, the separation constant, is equal to 0, -a^2, and a^2. But in this particular problem, instead of substituting λ=0, λ = a^2, λ= -a^2, they substitute the entire coefficient of X, (λ-1)/k =0, (λ-1)/k =a^2, and (λ-1)/k =-a^2. I've linked a picture of what it says in the solution manual. I don't understand why in this particular problem they substituted the entire coefficient of X rather than just λ.

http://img13.imageshack.us/img13/2751/91625740.jpg [Broken]

 

[Fredrik]

The equality f''(x)+cf(x)=0 defines a set of differential equations. It contains one equation for each value of c. You can use the same mathematical expression for the general solution of any equation in (any) one of the subsets defined by c<0, c=0 and c>0, but you need a different mathematical expression for each of those three subsets.

 

[djeitnstine]

So my book says that to solve a PDE by separation of variables, we check the three cases where λ, the separation constant, is equal to 0, -a^2, and a^2. But in this particular problem, instead of substituting λ=0, λ = a^2, λ= -a^2, they substitute the entire coefficient of X, (λ-1)/k =0, (λ-1)/k =a^2, and (λ-1)/k =-a^2. I've linked a picture of what it says in the solution manual. I don't understand why in this particular problem they substituted the entire coefficient of X rather than just λ.

http://img13.imageshack.us/img13/2751/91625740.jpg [Broken]

You should try to do the working for yourself. I don't have time to show you what they did in LaTex, but here is the nitty gritty

kX'' - X/X = -L

kX'' = -LX + X

X'' = -LX + X/k

X'' = X(-L+1/k)

X'' + X(-L+1/k) = 0