In this problem we find the eigenfunctions and eigenvalues of the differential equation
\frac{d^2y}{dx^2} + \lambda y = 0
with boundary conditions
y(0)+y^\prime(0) = 0 \hskip {20pt}
y (3) = 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.
y(x) = .

Apply the boundary conditions to the general solution to obtain two equations relatingA to B :

= 0

= 0

(1b.) Solving this system forA and B we get

A =

B =

** Case 2: ** \lambda < 0
y(x) = .

Apply the boundary conditions to obtain two equations relatingA to B :

= 0

= 0
\sqrt{-\lambda}=

(2c.) Choose the graph of\sqrt{-\lambda} and \tanh\left(3\sqrt{-\lambda}\right)

A

B

C

D

From the graphs there are negative eigenvalues \lambda . They are
\lambda = (Your answers should be accurate to at least 10 decimal places.)
y(x) = .
Apply the boundary conditions to get two equations relating A to B :
= 0

= 0
\sqrt{\lambda} =

In increasing order, the first four positive eigenvalues from this equation are:
\lambda =
(Your answers should be accurate to at least 8 decimal places.)

** Case 1: **

(1a.) Ignoring the boundary conditions for a moment, the general solution of the differential equation is

Apply the boundary conditions to the general solution to obtain two equations relating

(1b.) Solving this system for

A =

B =

(2a.) Ignoring the boundary conditions for a moment, the general solution is

Apply the boundary conditions to obtain two equations relating

(2b.) Assuming

(2c.) Choose the graph of

A | B | C | D |

From the graphs there are

Sturm-Liouville guarantees infinitely many eigenvalues, so we go on.

** Case 3: **

(3a.) Ignoring the boundary conditions for a moment, the general solution is

(3b.) Assuming

In increasing order, the first four positive eigenvalues from this equation are:

(4.) List the first three eigenfunctions in order of increasing eigenvalues, including functions with negative eigenvalues.

Here are graphs of the first four eigenfunctions on

List the functions, ordered by their eigenvalues in increasing order, by their color - green, red, blue, black:

You can earn partial credit on this problem.