We show how linear systems can be written in matrix form, and we make many comparisons to topics we have studied earlier.
Linear Systems of Differential Equations
A first order system of differential equations that can be written in the formis called a linear system.
The linear system (eq:10.2.1) can be written in matrix form as
or more briefly aswhere We call the coefficient matrix of (eq:10.2.2) and the forcing function. We’ll say that and are continuous if their entries are continuous. If , then (eq:10.2.2) is homogeneous; otherwise, (eq:10.2.2) is nonhomogeneous.
An initial value problem for (eq:10.2.2) consists of finding a solution of (eq:10.2.2) that equals a given constant vector at some initial point . We write this initial value problem as
The next theorem gives sufficient conditions for the existence of solutions of initial value problems for (eq:10.2.2). We omit the proof.
- Write the system in matrix form and conclude from Theorem thmtype:10.2.1 that every initial value problem for (eq:10.2.3) has a unique solution on .
- Verify that is a solution of (eq:10.2.3) for all values of the constants and .
- Find the solution of the initial value problem
item:10.2.1b If is given by (eq:10.2.4), then
item:10.2.1c We must choose and in (eq:10.2.4) so that which is equivalent to Solving this system yields , , so is the solution of (eq:10.2.5).
Trench, William F., ”Elementary Differential Equations” (2013). Faculty Authored and Edited Books & CDs. 8. (CC-BY-NC-SA)