# Solve Higher Order DE: Factoring to Find General Solution

• suspenc3
In summary, the conversation is about factoring a higher order differential equation using binomial form and Pascal's Triangle. The pattern is shown and the solution is found using the pattern.
suspenc3
When trying to solve a higher order differential equation, how can I factor it to find the general solution. A question like :$$y^{(4)}-4y^{(3)}+6y^{(2)}-4y^{(1)}+y=0$$.
So I can write this as:$$m^4-4m^3+6m^2-4y^1=0$$. How can I factor m? The prof mentioned something about Pascal's Triangle, but I'm not sure what to do, I haven't even looked at Pascals Triangle in I don't even know how many years.

Did you mistype something, please check it again.

Here is Pascal's Triangle.

http://en.wikipedia.org/wiki/Pascal's_triangle

The first equation you gave, in binomial form is: $$(y-1)^4=y^4(-1)^0+4y^3(-1)^1+6y^2(-1)^2+4y^1(-1)^3+y^0(-1)^4=y^4-4y^3+6y^2-4y+1$$

Do you see the pattern?

Last edited:
Yeessss, I suppose

(y+1)^0=1=1
(y+1)^1=y+1=1 1
(y+1)^2=y+2y^2+1=1 2 1
(y+1)^3=y^3+3y^2+3y+1=1 3 3 1

etc

Thanks Alot!

## 1. What is a higher order differential equation?

A higher order differential equation is a type of mathematical equation that involves derivatives of a function up to a certain order. For example, a second order differential equation involves the second derivative of a function, while a third order differential equation involves the third derivative, and so on.

## 2. Why is factoring used to solve higher order differential equations?

Factoring is used to solve higher order differential equations because it allows us to rewrite the equation in a simpler form, making it easier to solve. Factoring also helps us to identify any repeated roots or patterns in the equation, which can help in finding the general solution.

## 3. How do you factor a higher order differential equation?

To factor a higher order differential equation, you need to first identify the highest derivative in the equation. Then, you need to find the common factors of all the terms involving that derivative. These common factors can be factored out, leaving behind a simpler equation that can be solved to find the general solution.

## 4. What is the general solution of a higher order differential equation?

The general solution of a higher order differential equation is a family of functions that satisfies the given equation. It includes all possible solutions, and may contain arbitrary constants. To find the particular solution, we need to apply initial or boundary conditions to the general solution.

## 5. Are there any specific strategies or tips for factoring higher order differential equations?

One useful strategy for factoring higher order differential equations is to look for any patterns in the equation and try to factor it accordingly. Another tip is to check for any common factors or repeated roots, which can be factored out. It may also be helpful to practice factoring simpler equations first before attempting more complex ones.

• Precalculus Mathematics Homework Help
Replies
13
Views
797
• Precalculus Mathematics Homework Help
Replies
5
Views
331
• Precalculus Mathematics Homework Help
Replies
1
Views
1K
• Calculus and Beyond Homework Help
Replies
7
Views
862
• Precalculus Mathematics Homework Help
Replies
2
Views
1K
• Precalculus Mathematics Homework Help
Replies
4
Views
874
• Precalculus Mathematics Homework Help
Replies
2
Views
648
• Precalculus Mathematics Homework Help
Replies
8
Views
2K
• Precalculus Mathematics Homework Help
Replies
6
Views
1K
• Calculus and Beyond Homework Help
Replies
3
Views
704