# Finding Eigenvalues for u''+λu=0

• beetle2
In summary, eigenvalues are special numbers associated with a square matrix that represent the scaling factor of the corresponding eigenvector. They are important in solving systems of linear equations and understanding the behavior of dynamical systems. To find eigenvalues for a differential equation, we first rewrite the equation in matrix form and solve for the characteristic equation. The solutions of the characteristic equation are the eigenvalues. Eigenvalues are significant in solving differential equations because they allow us to find the general solution. Complex eigenvalues can also arise in solving differential equations with non-constant coefficients and have important implications in studying stability and behavior. A matrix can have repeated eigenvalues, which occurs when the characteristic equation has repeated roots and can complicate the process of finding the general solution
beetle2
Hi guys,

Can someone please explain how you find the eigenvalues of this type?

$u''+\lambda u =0$

or point me to some decent literature?

regards
Brendan

It's a second order, homogenous ODE, see if knowing the name helps you find the method. If not, I'll help more.

## 1. What are eigenvalues?

Eigenvalues are special numbers associated with a square matrix that represent the scaling factor of the corresponding eigenvector. They are important in solving systems of linear equations and understanding the behavior of dynamical systems.

## 2. How do you find eigenvalues for a differential equation?

To find eigenvalues for a differential equation, we first rewrite the equation in matrix form by setting the differential operator equal to a matrix with the corresponding coefficients. Then, we solve for the characteristic equation by setting the determinant of the matrix equal to 0. The solutions of the characteristic equation are the eigenvalues.

## 3. What is the significance of eigenvalues in solving differential equations?

Eigenvalues are important in solving differential equations because they allow us to find the general solution of the equation. By finding the eigenvalues and corresponding eigenvectors, we can construct a fundamental set of solutions that can be combined to form the general solution.

## 4. Can a differential equation have complex eigenvalues?

Yes, a differential equation can have complex eigenvalues. In fact, complex eigenvalues often arise in solving differential equations with non-constant coefficients. Complex eigenvalues also have important implications in studying the stability and behavior of solutions.

## 5. Can a matrix have repeated eigenvalues?

Yes, a matrix can have repeated eigenvalues. This occurs when the characteristic equation has repeated roots. In this case, there will be more than one eigenvector associated with the same eigenvalue. This can complicate the process of finding the general solution, but it is still possible to do so.

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