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

to form the characteristic equation

[math]\displaystyle{ {\lambda^n +a_{n-1}\lambda^{n-1}+\cdots+a_1\lambda+a_0 = 0} }[/math]

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}.}