In this problem we find the eigenfunctions and eigenvalues of the differential equation $$\frac{d^2y}{dx^2} + \lambda y = 0$$ on the interval $0\leq x \leq a$, where $a>0$, with boundary values $$y(0) = 0\hskip {20pt} y(a) = 0.$$

For the general solution of the differential equation in the following cases use A and B for your constants, for example $y = A\cos (x) + B\sin(x)$. For the variable $\lambda$ type the word lambda, otherwise treat it as you would any other variable.

Case 1: $\lambda = 0$

(1a.) (Fill all three answer blanks to receive credit.) Ignoring the boundary values for a moment, the general solution of differential equation is

Apply the boundary conditions to the general solution to obtain two equations relating $A$ to $B$:

(1b.) Solving for A and B we obtain
Case 2: $\lambda < 0$

(2a.) (Fill all three answer blanks to receive credit.) Ignoring the boundary values for a moment, the general solution of differential equation is

Apply the boundary conditions to obtain equations relating $A$ to $B$:

(2b.) Since $\lambda, a \neq 0$, the only solution of these equations is

Case 3: $\lambda > 0$

(3a.) (Fill all three answer blanks to receive credit.) Ignoring the boundary values, the general solution is

Apply the boundary conditions to obtain equations relating $A$ to $B$:

(3b.) Assuming the solution is not identically zero, these equations say that

(3c.) For all integers n, $\sin(n\pi) = 0$, so the eigenvalues are

for positive integers $n=1,2,3,4,\cdots$.

(3d.) List the first four nonzero eigenfunctions $y_1,y_2,y_3,y_4$ in order of increasing eigenvalue.
$y_1(x)=$, $y_2(x)=$, $y_3(x)=$, $y_4(x)=$,
Here are graphs of the first four nonzero eigenfunctions on $\lbrack 0,3 \rbrack$.

(3e.) List the eigenfunctions in increasing order of their eigenvalues by their color - green, red, blue, black: