Rayleigh Flow - Integration By Parts

[apennine]

Hello,

This is not a question regarding a homework problem, but a step in class the professor did not show how to calculate.

Homework Statement

I am taking a course on Viscous Flow, and for Rayleigh flow after applying the similiarity solution : $$\eta=(y/(2*\sqrt{\gamma*t}))$$

The x-momentum equation is given by $$f^{''}+2*\eta*f^{'}=0$$

He states that "after integrating by parts the following solution is obtained"

$$f=C_{1}*\int(exp(-\eta^{2})*d\eta)+C_{2}$$ (integral is from 0 to $$\eta)$$

I probably should, but I don't understand how to integrate the original governing equation by parts and obtain that solution. I have reviewed my old calculus books and cannot find any type of example which is similar. Can anyone shed any light?

 

[Dick]

Call g=f'. Then the ode is dg/d(eta)+2*eta*g=0. Now just solve for g by separation of variables. Finally integrate g to get f. I'm not sure I would call that 'integration by parts'.