Eigenvalues for a 4th order boundary value problem

[Xyius]

Homework Statement

$$y^{(4)}+\lambda y=0$$
$$y(0)=y'(0)=0$$
$$y(L)=y'(L)=0$$

Homework Equations

The hint says...
$$let \lambda = -\mu ^4, \mu >0 or \lambda = 0$$

The Attempt at a Solution

Listening to the hint, I got
$$r=\pm\mu$$ With multiplicity 2 of each. So that means..
$$y=c_1 e^{\mu t}+c_2te^{\mu t}+c_3e^{-\mu t}+c_4te^{-\mu t}$$
Finding the derivative and solving the system of the 4x4 matrix (Thank god for the TI-89!) I find that all constants are equal to zero. When lambda equals zero I also get zero for all the constants.

The answer in the back of the book says..
for $$\lambda = -\mu^4$$ and
$$cos(\mu_n L)cosh(\mu_n L)=1$$
$$y_n=c_n[(cos(\mu_n L)-cosh(\mu_n L))(sin(\mu_n x)-sinh(\mu_n x))-(sin(\mu_n L)-sinh(\mu_n L))(cos(\mu_n x)-cosh(\mu_n x))]$$

I do not understand where they got this. It looks like there are only 2 cases and each case has the trivial solution. Can anyone help me out?

 

[mighty2000]

It seems like you have found the general solution for the given differential equation, but the answer in the back of the book is providing a specific solution for the given boundary conditions. This solution is known as the eigenfunction expansion method and it is commonly used in solving boundary value problems.

To understand where the solution in the book is coming from, we first need to understand the concept of eigenfunctions and eigenvalues. In this case, the differential equation can be written as a second-order ODE with constant coefficients, which has the general form of y'' + \lambda y = 0. This type of equation has a set of special solutions called eigenfunctions, which are functions that satisfy the equation for certain values of \lambda, known as eigenvalues.

In our case, the eigenfunctions are of the form y_n(x) = cos(\mu_n x) and y_n(x) = sin(\mu_n x), where \mu_n is a positive constant. Plugging these functions into the differential equation, we get \lambda = -\mu_n^4, which is the eigenvalue corresponding to the eigenfunction y_n(x). This means that for each value of n, we have a different eigenfunction and eigenvalue.

Now, to find the specific solution for the given boundary conditions, we need to use the eigenfunction expansion method. This method states that any solution to the differential equation can be written as a linear combination of the eigenfunctions, with coefficients determined by the boundary conditions. In our case, we have four boundary conditions, so we need to use four eigenfunctions, which are given by y_n(x) = cos(\mu_n x) and y_n(x) = sin(\mu_n x) for n = 1,2,3,4.

Using these eigenfunctions, we can write the solution as y(x) = \sum_{n=1}^{4} c_n y_n(x). Now, we need to determine the coefficients c_n by plugging in the boundary conditions. For example, plugging in x=0 and y=0 gives us c_1 = 0, plugging in x=L and y=0 gives us c_2 = 0, and so on. This will give us a system of equations for the coefficients, which we can solve to get the specific solution provided in the book.

In summary, the solution in the book is a specific solution obtained using the eigenfunction expansion method, while