A simple diff. eq. question...

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Sarah B. - 23 Jul 2006 19:34 GMT

If there is a function f(u,v) such that df/dv = u (df^2 / dvdu), what is f?
Is there any kind soul out there that could enlighten me? :-)

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[Mr.] Lynn Kurtz - 23 Jul 2006 21:10 GMT

>If there is a function f(u,v) such that df/dv = u (df^2 / dvdu), what is f?
>Is there any kind soul out there that could enlighten me? :-)

How about f(u,v) = h(u) + u g(v)

where h and g are arbitrary differentiable functions?

--Lynn

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Omega Cubed - 23 Jul 2006 22:51 GMT

> If there is a function f(u,v) such that df/dv = u (df^2 / dvdu), what is f?
> Is there any kind soul out there that could enlighten me? :-)

Separation of variables: f(u,v) = g(u)h(v)

df/dv = g h'
df^2/dvdu = g'h'

so you have

gh' = u g'h'

so for any differentiable function h(v), multiply it by a solution of

g(u) = u g'(u)

The solution to that last ODE is an exercise for the reader.

--Ohm3

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Math Forum / Mathematics / Undergraduate Math / July 2006

A simple diff. eq. question...