Use the differential operator to solve this differential equation

[s3a]

Homework Statement

Use the differential operator method to find a fundamental set of solutions {y_1(x), y_2(x)} of the equation

d^2 y/dx^2 - 18 dy/dx + 90y = 0.

Homework Equations

Differential operator method.

The Attempt at a Solution

I have a huge problem understanding how to use the differential operator method. I can successfully complete this problem using the characteristic equation:

r^2 - 18r + 90 = 0
r = 9 +/- 3i
y_1(x) = e^(9x) * cos(3x)
y_2(x) = e^(9x) * sin(3x)

but I really need to understand how to use the differential operator and I don't get anything I read on the internet or in my textbook.

A comparison (with contrasting) to the method with the characteristic equation would be GREATLY appreciated!
Thanks in advance!

 

[jambaugh]

Since the equation is constant coefficient, the characteristic polynomial is also the differential operator form:

Your equation takes the form:
$[D^2 -18D +90]y = 0$, where $D = \frac{d}{dx}$
Factoring the characteristic polynomial becomes a factorization of the differential operator:
$(D-r_1)(D-r_2)y = 0$
you can then successively integrate:
Solve the first order equation: $(D-r_1)y=0$ and let $y_1$ be your general solution.
Then solve the first order equation $(D-r_2)y = y_1$ and let that solution be $y_2$

Then observer that $(D-r_1)\left\{ (D-r_2) y_2 \right\} = (D-r_1) {y_1} = 0$ so $y_2$ is a solution to your original equation. If you gave the most general solution in each step you have the most general solution to the original problem.