Reduction of Order

Reduction of order is a method of take differential equations of a higher order and reduce it so that you can use a solving method that only works on first order derivative (such as separable variables or integrating factor). It requires that you already know one solution to the differential equation.

To preform reduction of order creates an unknown function of x (or whatever the independent variable is) and multiplies it by the already known solution () to get a new solution (referred to as moving forward). below is the given solution.

Where is the unknown function. There derivative is then taken as to as high an order as the differential equation (ex. get and ). These are then substituted back into the original differential equation (and reduce).

Setting a new variable equal to the unknown function's derivative () we get:

We can this use appropriate solving methods to get , , and then finally . See W2L5 - Reduction of Order and Chapter 4.2 - Reduction of Order for more info and examples.