Solving a differential equation using Cauchy-Euler Method

[madcattle]

Homework Statement

Hey all, this is my first time posting here. I've used your help before, but have not actually ever had to post a question. Thanks for any help you can give, I am excited to finally join the community.

I am doing calculus homework and I am having trouble solving this problem using the Cauchy-Euler method that we're supposed to solve it with.

Solve: xy⁴ + 6y^m = 0

It is the part about the 6y^m that I am not sure what to do with.

Homework Equations

a_ndⁿy / dxⁿ ...a₀y = 0

The Attempt at a Solution

My book toldme to try and substitute y=x^m into the derivative parts, and all of the other problems I've done work with that method. This is what I've done so far.

y=x^m
y¹=mx^m-1
y²=m(m-1)x^m-2
y³=m(m-1)(m-2)x^m-3
y⁴=m(m-1)(m-2)(m-3)x^m-4

I know how to treat the 4th derivative term; again, I just don't know what or how to think about the y^m term. I know the solution to the problem, as it is in the back of my book, I just don't know how to get there. Can anyone give me a prod in the right direction?
Thanks

 

[HallsofIvy]

I don't see any derivative in the original equation!

 

[madcattle]

I'm sorry, our book uses notation so that y^4 is supposed to be the fourth derivative of y.
So, for example, y¹=y^´ and y^2 = y´´

 

[Ray Vickson]

I'm sorry, our book uses notation so that y^4 is supposed to be the fourth derivative of y.
So, for example, y¹=y^´ and y^2 = y´´

The usual notation is y^(2) (so we can tell the difference between that and the square of y). Anyway, you have found the first few y^(m) for y = x^n. If you don't see the pattern, I suggest you look instead at y^(m)/m! and see if you recognize the coefficients there.

RGV

 

[madcattle]

I see that there is a relationship in that
y^(m)/m = (my^(m-1) + m(m-1)y^(m-2)...)/(m(m-1)(m-2)...) but I don't actually understand how to apply that relationship at all yet

 

[madcattle]

Nevermind, I'm really sorry about this. I mistook the book for having written y^m when they really wrote y^´´´. I guess my new problem is poor eyesight. Thanks a lot for the help, I struggled trying to solve this for a solid 40 minutes.