Thursday, May 9, 2013

What the Wronskian? By Spencer S.

          Differential Equations is the name, insanely long math problems is the game. This semester’s post-calculus math elective, taught by Jeff Gadette, is a mind-bending, frustrating but rewarding lesson in the triumphs and tribulations (mostly tribulations) of applied and advanced uses of calculus. Differential Equations centers on the idea that you have some function, y(x), that combined with itself and its derivatives and other functions can give you a certain result. Given a jumble of y(x), y’(x), other functions of x, etc., all added or multiplied together or more, the goal is to find your one true function, y(x).
            When will we use this in real life, you may ask? Newton’s First Law of Thermodynamics can be written as a differential equation. Population growth of r- and k-strategist organisms can be modeled using a differential equation. Electrical engineers must use differential equations in their work. Students in the differential equations class must use differential equations to pass the class...
            Class time is usually taken up solving one or two problems. Don’t believe that a tiny math problem can take up to an hour? Look at this one.
2y’’ + y = tan(x)
            In English, that problem tells you that if you take the function y(x) and add it to two times its second derivative you end up with the tangent of x. Now, you need to find what equals y(x).
            If the problem didn’t have that tan(x), and instead 2y’’ + y = 0, the answer is , where c1 and c2 are arbitrary constants (any number you want). Don’t ask why. Now, by making 2y’’ + y = tan(x) instead of 0, here’s the answer, provided by Wolfram Alpha.


            Now you see the foundation of our frustrations. This question was on a test, and Gadette himself admitted it was impossible.

PS- The Wronskian of a set of functions is an n x n matrix consisting of the functions in the first row, their first derivative in the second row, their third derivative in the third row, and so on. This is useful in solving differential equations and it would take several weeks to explain why.

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