11 - Introduction to Reducible Second Order ODEs

THIS LECTURE IS OPTIONAL. YOU WILL MOST PROBABLY NOT NEED THIS FOR THE TEST, WE WILL ONLY BE DEALING WITH CONSTANT COEFFICIENT ODEs. You can skip this lecture if you feel like it.

What are these "reducible second order ordinary differential equations"?

Derivatives of higher order

Like we said, since derivatives have orders, so do differential equations.
In the context of differential equations, this order is the highest order derivative present in the equation.

What does "reducible" mean in this case?

This means that we can make a substitution. Some kind of variable that represents the first derivative. When we have this, we can represent the second derivative as the first derivative of this variable.

Effectively, we are "reducing" this differential equation to a first order one.

Two cases

There will be two cases when doing this type of solving. Either the variable x, or the variable y will be gone. We'll get into that in a bit.

Remember:

Since we have a second order ODE, we'll be "undoing" two differentiations. This means that we'll end up with two constants and .