# Eigenvalues, Eigenfunctions and Differential Equations

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Consider the equation: y'' + k*y = 0 with BC: y(0) = 0 , y() = a 0

Answer the following:

1. What are the restrictions on k such that there is a nontrivial solution?

2. Find a solution using eigenfunction expansion on [0,]

3. Find a solution to the differential equation using any method

4. Compare answers 2 and 3 show they are equivalent

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Eigenvalues, Eigenfunctions and Differential Equations are investigated.

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Consider the equation: y'' + k*y = 0 with BC: y(0) = 0 , y() = a 0

Answer the following:

1. What are the restrictions on k such that there is a nontrivial solution?

(1) k < 0. Let's write , where . Then the equation becomes and its general solution is . If we use the first boundary condition, y(0) = 0, we get . If we use the second condition y() = a, ...

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