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 3 with boundary values
y^\prime(0) = 0\hspace{20pt} y(3)+y^\prime(3) = 0
y(x) = .

= 0

= 0

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

A =

B =

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

= 0

= 0

(2b.) IfA and B are not both zero then, solving these two equations, one can eliminate both A and B and derive the following formula for \sqrt{-\lambda} in terms of hyperbolic functions (\cosh, \sinh, \tanh, \coth , etc.).
\sqrt{-\lambda} =

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

A

B

C

D

From the graph there are nonzero eigenvalues for \lambda ,
Sturm-Loiuville guarantees infinitely many eigenvalues, so we go on.
y(x) = .

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

= 0

= 0

(3b.) AssumingA and B are not both zero then, solving these two equations, one can eliminate both A and B and derive the following formula for \sqrt{\lambda} in terms of trigonometric functions.

\sqrt{\lambda} =

(3c.) In increasing order, the first four eigenvalues from this equation are:

\lambda =

(Your answers should be accurate to at least 7 decimal places.)

For the general solution of the differential equation in the following cases use A and B for your constants, for example

** Case 1: **

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

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

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

A =

B =

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

Apply the boundary conditions to obtain two equations relating

(2b.) If

(2c.) Choose the graph of

A | B | C | D |

From the graph there are

** Case 3: **

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

Apply the boundary conditions to obtain two equations relating

(3b.) Assuming

(3c.) In increasing order, the first four eigenvalues from this equation are:

(Your answers should be accurate to at least 7 decimal places.)

List the first three eigenfunctions

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.