Question

What's the answer to this differential equation if n is the nth derivative.

Answer 1

\[\LARGE xy^{(n+1)}=-ny^{(n)}\]

Answer 2

over 9000

Answer 3

I'm just curious to see what people do, I know one answer that will work because I made this up.

Answer 4

if you try different values for n, and use wolfram
you get

n y
0 A
1 A + B ln x
2 A + B ln x + Cx
3 A + B ln x + Cx + Dx^2

Answer 5

so \[y(x) = \sum^n_{i=0} a_i\frac{\mathrm d}{\mathrm dx}x^{i-1}\]

Answer 6

I guess this is more or less the same thing as what you said except I don't know how you're getting in the natural log with yours, just the indefinite integral of this with arbitrary constants a_n. \[\LARGE y(x)=\int\limits (\sum_{n=1}^\infty a_nx^{-n} )dx\]

Answer 7

hmm, i was kinda close

Answer 8

Until you wrote that the only answer I knew of that worked was

y=ln(x)

I came up with this trying to relate the differential equation y'=y which has the solution e^x to the differential equation that has the solution as its inverse, y=lnx.

Answer 9

the log term comes in at the start , when i=0, so x^{i-1} = x^{-1} d/dx x^{-1} = ln x

Answer 10

I want to try to find a way to take a differential equation and rewrite it in terms of its inverse function in hopes that I can simplify hard differential equations. After all, a differential equation is just relating a function to its slopes and concavity which inverse functions already have relations going on there. I just don't know how to do it yet lol.

@UnkleRhaukus You're right, then we get the constant of integration since it is simply an indefinite integral to complete all the terms.