## Elementary Differential Equations – Rainville

I finished the first two chapters of Elementary Differential Equations by Rainville.  This is the second edition circa 1958, and there is an 8th edition now.  This was a very through start into ODEs.  I wouldn’t have understood most of it without the MIT lectures.  This is my Dad’s book!  A couple of specific notes:

Chapter 1 – it is possible to move from a solution with $n$ arbitrary constants back to the differential equation, by differentiating and solving for each of the constants and substituting back in.  $n$ differentiations for $n$ constants.

Chapter 2 – homogeneous functions have all terms to the same degree.  The substitution could be $v = x/y$ or $v = y/x$ whatever is simpler.   Pg. 29 discusses exact equations where are of the form

$M(x,y) dx + N(x,y) dy = 0$

where ${ \partial M \over \partial x} = {\partial N \over \partial y}$.  If an equation is not exact it might be made so by an integrating factor.