# Liner differentials of order n, Kernel

• bakin
In summary, the given function y(x) = x-2 is in the kernel of L, where L = x^2D^2 + 2xD - 2, as verified by plugging in the first and second derivatives of y(x) and simplifying to get L(y) = 0.
bakin

## Homework Statement

Verify that the given function is in the kernel of L.

y(x)=x-2
L = x2D2 + 2xD - 2

## The Attempt at a Solution

I took the first and 2nd derivative of y(x), and got
y'(x)= -2x-3
y''(x)= 6x-4

Then plugged it into L (and a little simplifying) and got

L(y) = 6x-2+2x-1-2

I think I'm supposed to plug it in, and verify that it's equal to zero, but it's not coming out right.

Any obvious mistakes? Or wrong direction all together?

How does 2xDy become 2x^(-1)?? And the -2 isn't just a -2. L is operating on y. What should it be?

Bah, forgot about the -2 part. It's actually -2y, correct? So the last term would be -2y, or -2x-2.

And as I was typing out how I came up with 2xD, I realized I substituted just y into D, and not y'

With the correct substitutions, I came up with:

L = x2*6x-2 + 2x*-2x-3 - 2x-2

= 6x-2 - 4x-2 - 2x-2

= 0

:tongue: Once again, thanks Dick.

## 1. What is a linear differential of order n?

A linear differential of order n is a type of differential equation where the dependent variable and its derivatives are raised to the first power and are only multiplied by constants. It is considered to be a linear equation because the dependent variable and its derivatives do not appear in any other form, such as squares or products.

## 2. What is the kernel of a linear differential equation?

The kernel of a linear differential equation is the set of all solutions to the equation that have a derivative of zero. In other words, the kernel is the set of all constant solutions to the equation.

## 3. What does the order of a linear differential equation represent?

The order of a linear differential equation represents the highest derivative present in the equation. For example, a second-order linear differential equation will have the highest derivative as the second derivative.

## 4. How does one solve a linear differential equation of order n?

To solve a linear differential equation of order n, one must first rearrange the equation into standard form by grouping all the terms with the same derivative together. Then, the equation can be solved by using standard techniques such as separation of variables or the method of undetermined coefficients.

## 5. What is the difference between a linear differential equation and a non-linear differential equation?

The main difference between a linear differential equation and a non-linear differential equation is the form of the dependent variable and its derivatives. In a linear equation, the dependent variable and its derivatives are raised to the first power and are only multiplied by constants, while in a non-linear equation, they can appear in different forms such as squares or products. This leads to different methods of solving the equations.

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