# Solving a System Equation

### Why do we need to solve system equations?

Often during a course you will need to be able to solve a system equation for its roots. These roots can be complex, distinct, or repeated. These problems usually arise when working with linear systems or differential equations. A system equation is formatted as follows:

### System Equation:

### How to solve a system equation

For example purposes, I will solve a system equation with complex roots. A system equation with complex roots as a function of will appear in the following format (if it does not, you need to manipulate your equation to be in the form):

**Roots**:

So we have

which also equals

so your first step is to look at your equation and determine your roots, then write out your equation with constants.

**Example** with initial conditions and

so now we can write our equation as follows:

In order to solve for and we need to use our initial conditions. To evaluate the first derivative initial condition, we must first take the derivative of our that we just found.

evaluating this equation with t = 0 and the response equal to -4v, we get this:

evaluating our equation with t = 0 and the response equal to 3v, we calculate

Using these two equations, we calculate our constants:

and

Fill these into our equation to determine the final result.

Now you know how to solve this common differential equations and linear systems problem, determine characteristic roots and modes, and write system equations.