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

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July 26

Differential equation

What are the solutions of $(f\circ g{)}^{\prime }=g\circ f^{\prime }$? 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 = e^xs–ys. Expand the exponential: $0=\sum _{k=0}^{\mathrm{\infty }}{a}_k{s}^k=1+(x-y)s+\sum _{k=2}^{\mathrm{\infty }}\frac{x^k}{k!}{s}^k$. 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 $f(s)=\sum _{k=0}^{\mathrm{\infty }}{a}_k{s}^k$ 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).