Distinguish between methods

[Hyunmin Jung]

I can solve the problems but I have trouble distinguishing between methods. 2.1 #38. Why is this method called a variation of parameters? Is "Integrating Factor  where adding mu to the equation" a part of variation of parameters (Since you are technically adding a parameter to  p(t))? I always thought variation of parameters as adding some coefficient on complementary equation to find other fundamental solution of y to derive general solutions.

[Victor Ivrii]

I fixed you post full of unneeded line breaks and empty lines

Method of integrating factor: could be applied to first order equation $M(x,y)dx+N(x,y)dy=0$ which we multiply by $\mu$ to make it exact.

Method of variation of parameters could be applied to linear inhomogeneous first and higher order equations. See sections 2.1 and 3.6 (and later 4.4 and 7.9). For homogeneous equation solution is $y=C_1y_1+C_2y_2+\ldots+C_n y_n$ with constants $C_1,\ldots, C_n$ and for inhomogeneous equation solution is searched in the form  $y=C_1y_1+C_2y_2+\ldots+C_n y_n$ with unknown functions $C_1,\ldots, C_n$.

For first order linear equations these methods are equivalent.