Boundary conditions - unique solution

[aaaa202]

I recently solved a differential equation with the solution:

f(x) = Aexp(ikx) + Bexp(-ikx)

with the periodic boundary condition f(x+L)=f(x). This condition leads to:

Aexp(ikx)exp(ikL) + Bexp(-ikx)exp(-ikL) = Aexp(ikx) + Bexp(-ikx) (1)

Now the way I figured out the constants A and B was that I said, that (1) must be true in particular for x=0 and x=π/2k which gave some equations to solve for A and B.

But something appears weird to me with this method of using two arbitrary values for x in (1) to determine A and B. Namely how do I know that I should also get this value for A and B if I use two different values of x? I do realize you could maybe show this by a calculation letting x1 and x2 be arbitrary but from a bigger perspective: Is it always true that the solution of a differential equation is uniquely specified by a periodic boundary condition? Maybe I'm just confused about something simple - it's late at night..

 

[Chestermiller]

Do you require f(x) to be real, or are you also interested in cases where f(x) is complex?

Chet

 

[aaaa202]

I have no requirement on f being real.

 

[Chestermiller]

Actually, your periodicity requirement puts a restriction on k, not A and B: k = 2nπ/L, where n is any integer.

Chet

 

[blue_raver22]

I can understand your confusion and concerns about using two arbitrary values to determine the constants A and B in the solution of a differential equation with a periodic boundary condition. However, the fact that the solution (1) satisfies the given boundary condition for any value of x is a result of the uniqueness of the solution.

In mathematics, a unique solution refers to a solution that is the only possible outcome of a given set of conditions. In the case of a differential equation with a periodic boundary condition, the solution (1) is the only possible solution that satisfies the given condition. This is known as the uniqueness theorem for differential equations.

To understand this concept better, let's consider a simpler example. Suppose we have the equation x + y = 5 with the boundary condition y = 2. The unique solution to this equation is x = 3, which is the only possible value that satisfies both the equation and the boundary condition. Similarly, in your case, the solution (1) is the only possible solution that satisfies both the differential equation and the periodic boundary condition.

In general, the uniqueness of a solution depends on the boundary conditions and the properties of the differential equation. In some cases, the solution may not be unique, and in those cases, additional information or conditions may be needed to determine the constants.

In conclusion, the solution of a differential equation with a periodic boundary condition is uniquely specified, and using two arbitrary values to determine the constants is a valid approach. However, it is always important to check the properties and assumptions of the differential equation to ensure the uniqueness of the solution. I hope this explanation helps to clarify your confusion.