# Koeris/Notebook/2006-12-30

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 $\displaystyle{ \overrightarrow{v1},\overrightarrow{v2} }$. The equation then has the general form of a homogenous solution as follows: $\displaystyle{ y &= e^{\lambda x} }$

to form the characteristic equation

$\displaystyle{ {\lambda^n +a_{n-1}\lambda^{n-1}+\cdots+a_1\lambda+a_0 = 0} }$

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}.}