Differential Equations: The Epic Journey: Super serious and not so funny linear equations.

Section 2.4 is entitled “Linear Equations.”

A first order linear equation form:

If f(t) = 0, then this form is called homogeneous. Otherwise, it is called inhomogeneous. The functions f(t) and a(t) are the coefficients of the equation. Sometimes we can consider this equation in an even more general form:

This is still linear because you can just divide the b(t) over to put x prime by itself.

Something really important to note about linear equations is that x and x prime appear alone and only to the first order. Note: “…we do not allow x², (x^’)³, xx^’, e^x, cos(x^’) or anything for complicated than just x and x^’ to appear in the equation” (47).

Stay away, tricky calculus.

Kind of as a summary, here’s a small list of what’s considered linear and what’s not:

Linear equations can be solved exactly. Let us start with a general equation and show this:

So, when solving homogeneous equations for a general solution, there’s a basic process you can follow:

1. Separate variables

2. Integrate, rewrite constants as other constants if necessary, etc.

3. Go celebrate because you’re done

There are a couple more steps when solving an inhomogeneous equation.

So here are the steps:

0. For these steps, I’m referencing the following equation:

1. Rewrite as

2. Multiply by an integrating factor:

3. Integrate:

4. Solve for x(t).

Here’s an example (because the whole integrating factor thing should make more sense once you’ve seen an example):

Multiply by the integrating factor:

There’s also an alternate solution method for solving these equations, and I’m going to use a pretty generic example to show how it’s done.

The function v that we used to substitute is unknown (obviously). It is sometimes called a variable parameter, and therefore the method is called variation of parameters.

Now, the steps:

0. For these steps, I’m referencing the following equation: