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Example of Differential Equation

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In algebra, one uses certain tricks to find the values of a variable called "x".  In differential equations, one uses certain tricks to find the values of a function called "f(x)".
What used to drive me crazy were the proofs.  Why do they go through the proofs in class, when they're already in the book?  It'd be better using that time on practicing the solution to more of the problems!

Here we have a problem, and we should have an inkling of the concept.  It's a first order differential, which suggests an integrating factor.  There is also a condition for finding the constant of integration.
So, using the advantage of the exponential, we integrate the coefficient of the function to find the function.  Note that I even proved that the integration factor works for this problem, although it is really not necessary.
Other concepts needed from lower math: what the cotangent is, integration by substitution, calculus, and algebra.


The next part resets the equation, then integrates both sides.  Don't forget to multiply BOTH sides by the integrating factor and to divide BOTH sides by the coefficient function.  It helps to know trigometric functions, but substitute sine and cosine to be sure.
Once we get the function, the next thing is to solve for the constant.  Now any constant will suffice in the check on the answer, so we must find this particular constant now.

Once we have the constant of integration solved, then we can check the answer.  Obviously, we must know the quotient rule, for this function is a fraction.  If necessary, look it up.  Knowing what to do is more important than how to do it.
Simplify the derivative.  Then substitute into the original equation, just like algebra.
Note that ideally, the substitution should cancel all the terms, save the one related to the answer in the original equation.  It is better to know what the answer should be, so that any errors in getting there can be ironed out.  I took a few times on this problem recently because I knew what the answer should be.  Practice!


The professor wants you to prove you know the concepts.  To avoid the silly mistakes while getting there, you MUST practice the problems, often too few given.  In reality, there is truth to the axiom that one should spend two hours on the subject for every hour in class.