What makes uiv = u a 4th-order ODE?

[bravelittlemu]

I am not interested in the solution, but I am curious what makes: u^iv = u, a 4th-order ordinary differential equation. 'i' is the square-root of -1, v is some element of the reals, and the differentiating variable is x.


cheers

 

[MaxL]

what makes: u^iv = u, a 4th-order ordinary differential equation.

Really interesting! Thanks for bringing this to my attention!

I'm not familiar with this problem, so I'm not offering an answer, but here's what I did:

First restate

u^i*v = e^(ln(u)*i*v)=u

If you take the x derivative of both sides, you get

i*e^(ln(u)*iv)*(u'v/u + v'ln(u))=u'

recognize that the first part is u by the original eqn, then divide both sides by u',

iv +iv' u/u' = 1

That guy is separable, so

v'/(1-iv) = u'/(u*ln(u)) = a set of constants

Looks like a first order equation to me.

But I'm wondering if maybe that i allows higher orders? Like the way that i^i = (-e^(i*2*pi*n)*i/2) = -e^(pi*n) for any integer value of n? But that trick only works because i has an absolute value of 1, which isn't necessarily true of u or v. Also, I have no idea why n would stop at 4.

Where did you hear it was a 4th order ODE?

 

[bravelittlemu]

My textbook. :D
Birkhoff, Rota, Ordinary Differnential Equations (4th), 1989, pg 73 Example 1

Really interesting! Thanks for bringing this to my attention!
Where did you hear it was a 4th order ODE?

 

[hamster143]

This is not a differential equation at all. It's simply

u^{iv-1} = 1

(iv-1) \ln (re^{i\phi}) = 0

(iv-1) (i\phi + \ln{r}) = 0[/itex]<br /> <br /> which has solutions: (r=1, phi=0 =&gt; u=1) when v!=0, and any u otherwise.

 

[bravelittlemu]

This is not a differential equation at all.

Hi Hamster,

My textbook explicitly states that it is a 4th-order DE and the purpose of the example is to build a basis of solutions to the DE.

 

[hamster143]

The only way it could be a DE is if 'iv' denotes fourth derivative (Roman "4") rather than i times v:

u&#039;&#039;&#039;&#039; = u

 

[bravelittlemu]

The only way it could be a DE is if 'iv' denotes a fourth differential rather than i times v:

u&#039;&#039;&#039;&#039; = u

That is it. The book uses z = μ + iυ (nu) to represent a complex number and upon closer inspection (holding the book up to my face) the exponent is iv (in cursive, go figure.).

Thanks all!

 

[MaxL]

Hahaha, oh man that's hilarious.

 

[torquil]