For a freshman calculus derivation of this, see my book The Science of Radio, 2nd edition, Springer-Verlag 2001, pp. 416–418.
|2.||The problem quotation is from Larry C. Andrews, Introduction to Differential Equations with Boundary Value Problems, HarperCollins 1991, Problem 22, p. 91. This problem is actually a favorite of textbook writers (recall the quotation from Smith (1917) in What You Need to Know …), and the very same problem — with the piloted plane replaced with a bird — can be found in a book by two of my former colleagues at Harvey Mudd College, the mathematicians R. L. Borrelli and C. S. Coleman, Differential Equations: A Modeling Approach, Prentice-Hall 1987, pp. 112–115. You can find it again in the older (but still outstanding) text by Ralph Palmer Agnew, Differential Equations, McGraw-Hill 1960, pp. 146–147. I bought my copy of Agnew new, when a sophomore at Stanford, and it is now — nearly fifty years later — tattered and torn for all the use I’ve given it over the last five decades.|
|3.||This problem has a superficial resemblance to the much more difficult problem of determining the path that minimizes the time required to cross the river. That problem was solved in 1931 by the German mathematician Ernst Zermelo (1871–1953) — and so is called Zermelo’s problem in the mathematical literature — and it requires mathematics (calculus of variations) beyond the level of this book. You can find an extended discussion of that calculus in my book When Least is Best, Princeton 2004, pp. 200–278.|