Find two real functions whose sum is the ratio of their first derivatives, i.e. solve:

Or prove no solutions exist.

(This is a problem that cropped up in another thread: http://www.mathhelpforum.com/math-he...tml#post345253)

Results 1 to 3 of 3

- August 2nd 2009, 07:10 AM #1

- Joined
- Apr 2009
- From
- Atlanta, GA
- Posts
- 409

## A+b=a'/b'

Find two real functions whose sum is the ratio of their first derivatives, i.e. solve:

Or prove no solutions exist.

(This is a problem that cropped up in another thread: http://www.mathhelpforum.com/math-he...tml#post345253)

- August 2nd 2009, 02:14 PM #2

- Joined
- Jun 2009
- Posts
- 668
- Thanks
- 135

There are an infinite number of solutions aren't there ?

Just solve as a first order ode in .

Rewrite as

,

multiply by the integrating factor

and the equation can be written as

so

and

.

Now just choose something convenient for .

An easy choice would be after which an integration by parts would lead to .

- August 3rd 2009, 02:07 PM #3