# Homework Help: Differential equation with only the trivial solution

1. Dec 27, 2012

### Bipolarity

1. The problem statement, all variables and given/known data
Find a differential equation with its only (complex-valued) solution being y=0

2. Relevant equations

3. The attempt at a solution
I believe that there is no DE having only y=0 as its solution, but frankly I am not sure if this is the case. I would like to know whether or not this is true, so that I know in which direction I can begin working my proof (or at least a hint is appreciated).

Also, something (in my mind) tells me this problem may have some connection with Wronskian matrices, but I have no clue really.

EDIT: I tried playing with some random diff Eqs, it seems that there are in fact trivial things like y+y'-y'=0, but this type of answer seems to be trivial to be of substance. Is there a DE that does not reduce to a trivial y=0?

Thanks!

BiP

2. Dec 27, 2012

### haruspex

One approach is to think of a function of y' that cannot be zero, f(y') say, then write f(y')y = 0.

3. Dec 27, 2012

### Dick

Well, f(y')=1 works. Still seems like kind of a cheat. Notice they also said complex valued. So something like f(y')=1+(y')^2 won't work either. Not to say you can't cook one up. Like f(y')=(1+(y')(y'*)). I'm just wondering if there is something nontrivial here.

Last edited: Dec 27, 2012
4. Dec 28, 2012

### haruspex

I was thinking of ey'y=0

5. Dec 28, 2012

### Bipolarity

Good work!!! And thanks!!!

BiP