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)

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- Aug 2nd 2009, 06:10 AM #1

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## 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)

- Aug 2nd 2009, 01:14 PM #2

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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 .

- Aug 3rd 2009, 01:07 PM #3