In this lecture we will begin studying a few methods for the solution of linear differential equations. In particular we will focus on those differential equations that arise from modeling linear mechanical systems such as a mass-spring system. First consider the following differential equation:

In order to solve this equation for a unique y(t), we need the initial conditions: . Let us start by writing this equation in state space form. Define . This gives us:

With initial conditions: . In matrix notation we have

This fits the general form:

with

Before we develop the complete solution for this general problem we will look at the solution of a very simple case. Suppose x(t) and u(t) are scalars. This would give us A = a and B = b also scalars. Consider the first order differential equation

We operate on this equation as follows

multiply both sides by

and integrate the expression with respect to time

This last equation is known as the scalar form of the variation of parameters formula. The components of this equation are as follows:

Example: Calculating the solution to a specific case of the scalar problem given above.

with input

Using the variation of parameters formula above we find: