Trouble simplifying a tough equation

Question

I'm having trouble simplifying the following equation. I've tried grouping terms in different ways, but it's not looking any more joyful. Can someone please help with its resolution? Hopefully by midnight?? $$ \omega - \ln Y=\ln\left(H p^2 a + \exp(ra)\right) - \ln N $$

What do you want to do with the equation ? – Amr Dec 31 '12 at 23:38 — Amr, Dec 31 '12 at 23:38
OH. Now I see (after looking at the answers)! – Amr Jan 01 '13 at 00:13 — Amr, Jan 01 '13 at 00:13

Adi Dani · Accepted Answer · 2013-04-30T20:00:19.027

5

$$ \omega - \ln Y=\ln\left(H p^2 a + \exp(ra)\right) - \ln N $$ $$ \omega =\ln\left(H p^2 a + \exp(ra)\right)+ \ln Y - \ln N $$ $$ \omega =\ln\left(H p^2 a + \exp(ra)\right)+ \ln (Y/N ) $$ $$ \omega =\ln\left({YH p^2 a + Y\exp(ra)\over N}\right) $$ $$ \omega =\ln\left({Ha p^2Y + Ye^{ar})\over N}\right) $$ $$e^{\omega}=\frac{Ha p^2Y + Ye^{ar}}{N}$$ $$0=Ha ppY-Ne^{\omega} + Ye^{ar}$$

edited Apr 30 '13 at 20:00

answered Jan 01 '13 at 00:02

Adi Dani

16,949

Or just $HappY = Ne^{\omega} - Ye^{ar}$ :). – mjqxxxx Jan 01 '13 at 00:14
O.K. THats sounds better – Adi Dani Jan 01 '13 at 00:16

c.p. · Answer 2 · 2013-01-01T00:03:57.030

3

Assuming everything there are numbers (or at least, everything commutes), may be you are wishing the community "$HappY Ne^\omega Ye^{ar}$".

edited Jan 01 '13 at 00:03

answered Dec 31 '12 at 23:47

c.p.

780

1

$1=HappY(Ne^\omega)^{-1}Ye^{ar}$ to be more accurate. – c.p. Jan 01 '13 at 00:01

score 1 · Answer 3 · answered Jan 01 '13 at 01:03

1

Here, I solved for $a$ for you:

$$ a = \frac{N r e^\omega-H p^2 Y \cdot \mathrm{W}\left(\frac{e^{\frac{e^\omega N r}{H p^2 Y}} r}{H p^2}\right)}{H p^2 r Y} $$

Where $\mathrm{W}(x)$ is the Lambert-W function.

answered Jan 01 '13 at 01:03

George V. Williams

5,213

Technically correct. – Thomas Jan 01 '13 at 03:55

Trouble simplifying a tough equation

3 Answers3