For some research Im doing, I've derived an equation of the form below for C...

Willow Pratt

Willow Pratt

Answered

2022-07-10

For some research Im doing, I've derived an equation of the form below for C ( r )
C + 2 r C = W + f C
Or, if you prefer,
C C + 2 r C C W C = f
This has the form of a 2nd order inhomogeneous linear equation with variable coefficients but the problem for me here is I don't know a clever way to solve this as C appears throughout equation. Any ideas on how to solve this and indeed, if there is even a solution ?
Id plug it into mathematica but am out of office travelling so would be most grateful for input!

Answer & Explanation

Jordan Mcpherson

Jordan Mcpherson

Expert

2022-07-11Added 16 answers

Hint:
Let U = r C,
Then d U d r = r d C d r + C
d 2 U d r 2 = r d 2 C d r 2 + d C d r + d C d r = r d 2 C d r 2 + 2 d C d r
1 r d 2 U d r 2 = W + f r U
d 2 U d r 2 = W r + f r 2 U
You can consider as two members Emden-Fowler type nonlinear ODE

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