# Koeris/Notebook/2006-12-30

### From OpenWetWare

Studying for the math quals... ODEs and PDEs

There are only a limited number of types of questions on the math qualifier exam. One that always crops up is solving a system of linear equations, either homogenous or non-homogenous, and usually with constant coefficients.

The approach to solving that is of course finding the determinant of the coefficient matrix and then using the eigenvalues to construct the eigenvectors . The equation then has the general form of a homogenous solution as follows:
**Failed to parse (syntax error): y &= e^{\lambda x}**

to form the characteristic equation

to obtain the solutions

\lambda=s_0, s_1, \dots, s_{n-1}.

When this polynomial has distinct roots, we have immediately n solutions to the differential equation in the form

{y_i(x)=e^{s_i x}.}