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.