Question

show that the one-parameter family of straight lines u = Ct + f(C) is a solution to the diff. eq. tu' - u - f(u') = 0 for any value of the constant C.

Answer 1

\(tu'-u-f(u')=tC-(Ct+f(C))+f(C)=0\)

Answer 2

what is the reasoning behind this?

Answer 3

For \(u=Ct+f(C)\) we have \(u'=C\) and you just plug in \(u\) and \(u'\) into the equation and verify that it is equal to zero.