Wikipedia:Reference desk/Archives/Mathematics/2015 July 26

= July 26 =

Differential equation
What are the solutions of $$(f \circ g)' = g \circ f'$$? Examples include f arbitrary and g(x) = x, and f(x) = x2 and g = sin. GeoffreyT2000 (talk) 01:33, 26 July 2015 (UTC)

an obscure root-finding problem
I expect the answer is No, but anyway: if $$y=r e^{x/r}$$, is there a closed expression for r given x,y? —Tamfang (talk) 08:40, 26 July 2015 (UTC)
 * There is! But it requires use of the non-elementary Lambert W function: $$r=\frac{-x}{W(-x/y)}$$. -- Meni Rosenfeld (talk) 09:15, 26 July 2015 (UTC)
 * Thanks! Looks like I need to install SciPy. —Tamfang (talk) 19:56, 26 July 2015 (UTC)
 * ...or work out the Taylor series (with y as the variable, in my application). —Tamfang (talk) 21:03, 27 July 2015 (UTC)

A closed expression is not necessarily useful. Set s = r–1. Then the equation is 0 = exs–ys. Expand the exponential: $$0 =\sum_{k=0}^\infty a_k s^k=1+(x-y)s+\sum_{k=2}^\infty {x^k \over k!}s^k$$. Leave it here until you need numeric solutions. Bo Jacoby (talk) 22:30, 26 July 2015 (UTC).
 * Numeric solutions are, as it happens, all I need. —Tamfang (talk) 21:03, 27 July 2015 (UTC)
 * If that. It hit me last night that the idea for which I asked this question was backward. Is that a valid reason to mark it Resolved? —Tamfang (talk) 06:14, 29 July 2015 (UTC)

Let $$f(s)=\sum_{k=0}^\infty a_k s^k$$ and $$f_N(s)=\sum_{k=0}^N a_k s^k$$. A root s in the algebraic equation $$0= f_N(s)$$ approximately solves the transcendental equation $$0= f(s)$$ when N is not too small. If you supply values for x and y I'll show you how to solve it. Bo Jacoby (talk) 06:54, 28 July 2015 (UTC).