Integrating the solar wind equation

[Jdraper]

Hi, eveyone have been struggling to do this problem for a long time now, figured it is something very simple I am missing so thought I should ask here.

1. Homework Statement
Parker's solar wind equation is given after some manipulation as:

(v - (C_s² / v) dv/dr = 2 (C_s² / r²) (r - r_c )


where r_c = GMm/4kT (the critical radius)
and
C_s² = 2kT/m (coronal sound speed)

The Attempt at a Solution

So using the usual method I separate the variables and integrate this equation to get:

v² / 2 - C_s² log(v) = 2C_s² log(r) + (4C_s² r_c / r) + C

where C is the constant of integration. However every solution

However every solution I see around the internet and indeed in my notes quotes the solution:

(v² / C_s²) - log(v²/ C_s²) = 4log(r/r_c) + 4(r/r_c)

and I cannot seem to figure out why. Am I missing something blindingly obvious here?

Thanks for your time, John.

 

[haruspex]

In your integration, the 4 should be 2.
You can get your version a lot closer by mutiplying through by 2/C_s².
The v² inside the log comes from bringing in the factor of 2 from outside, and the /C_s² may come from your constant of integration. (Likewise the /r_c in the other log.)
Theonly discrepancy that leaves is r/r_c versus r_c/r. Are you sure you have quoted that correctly?