Solve Higher Order DE: Factoring to Find General Solution

[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.

 

[rocomath]

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?

 

[suspenc3]

Yeessss, I suppose

 

[rocomath]

(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

 

[suspenc3]

Thanks Alot!