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.

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