# Use the differential operator to solve this differential equation

• s3a
In summary, the differential operator method is used to find a fundamental set of solutions for a constant coefficient equation. By factoring the characteristic polynomial into a factorization of the differential operator, you can successively integrate to find the general solutions. This method can be compared to the characteristic equation method, which also yields a fundamental set of solutions.
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!

Last edited:
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.

## What is a differential equation?

A differential equation is a mathematical equation that relates the values of an unknown function to its derivatives. It is commonly used to model physical phenomena in various fields such as physics, engineering, and economics.

## What is a differential operator?

A differential operator is a mathematical notation that represents the operation of taking a derivative. It is usually denoted as D or ∂ and is used to solve differential equations.

## How do you use the differential operator to solve a differential equation?

To use the differential operator to solve a differential equation, you first need to rewrite the equation in terms of the operator and the function. Then, you can apply the operator to both sides of the equation to isolate the function and solve for it.

## What are the common types of differential equations?

The common types of differential equations include linear and nonlinear equations, ordinary and partial differential equations, and first-order and higher-order equations. Each type has its own characteristics and methods for solving.

## What are the applications of solving differential equations?

Solving differential equations is essential in many scientific fields, such as physics, engineering, and biology. It is also used in various real-life applications, such as predicting the stock market, modeling population growth, and analyzing heat transfer.

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