**Linear First Order ODEs**

**Step 1: take derivative of (1),**

**Step 3: From (1), get x _{2} in terms
of x_{1}**

**Step 4: Substitute (5) into (4) and rearrange**

Thus the characteristic equation will be given by

Note: If we had tried to find a second order differential
equation in terms of x_{2} using the same procedure of substitution,
we will obtain

which will yield the same characteristic equation. Thus
there is really just one characteristic equation for the system.

The eigenvalues will just be the roots of the characteristic
equation.

**Example:**

Then the characteristic equation and the eigenvalues are:

**Step 2: Extract the matrix operator,**

**Step 2: Use Cramer’s rule,**

or,

which is the same result as Method 1 when D is replaced
back by d/dt.

can be obtained as

**Example:**

The characteristic equation and the eigenvalues are given
by

The system is unstable.

This page is maintained by Tomas B. Co (tbco@mtu.edu). Last revised 12/10/1999.

Tomas B. Co

Associate Professor

Department of Chemical Engineering

Michigan Technological University

1400 Townsend Avenue

Houghton, MI 49931-1295