MATH 2243 Lecture Notes - Lecture 1: Stochastic Process, Complex Conjugate, Indeterminate Form
Document Summary
This should be true for all values of t. We need to find functions that differentiate to themselves. There is a parameter r, which we need to determine. This is true for all values of t if. This is the characteristic equation, and there are 3 cases. *in all cases, we want real-valued functions as solutions. These will from a fundamental set of solutions. If we can solve for c1 and c2, then {y1,y2} is a fundamental set of solutions. Since r2 r1, (we assumed this) we have a solution to the ivp. This is 2nd order, linear, constant coefficient, homogeneous: if all of those things are true, we are allowed to assume. Create a new solution since {y1,y2} are a fundamental set of solutions. Note this form was chosen since as r1 r2, we get an indeterminate form! We do have a fundamental set of solutions: From the characteristic equation, fundamental set of solutions are.
