PREPARATION AND
DIFFERENTIATION OF SOLUTIONS
Answers
Answer:
Explanation:
The single best preparation for differential equations going is to take every derivative you find easy to solve and try to pose it as a question about the relationship between a function and its own derivatives.
So for instance, take the ideal analytic function f(x)=ex. This thing is bizarre, its derivative is also its integral (up to a constant) and both are equal to f(x).
So you ask yourself, okay what function has the property that ddxf(x)−∫f(x).dx=0 ?
Now vary the exponent. Stick some g(x) in there, say.
f(x)=eg(x)
Take derivatives, play with the easiest analytic functions and get a sense of the behaviours of these things.
You will want to feel comfortable with all the basic rational functions, their derivatives and integrals. Not expert, this is play, but enough to feel like sin, cosine, tangent, all their inverse functions, exponentials, logs are familar and their integrals and derivatives are familiar. Keep your eye on the simplest rational functions, they are the most commonly appearing functions in the solution of DE's.
Nothing will beat familiarity with the integrals and derivatives of the rational functions and small variations on that, when keeping up with the proofs in class. Nothing will beat that in terms of tackling problems knowing you have a solid background.
A lot of the class time will be spent learning methods which are there to kind of avoid needing to use any intuition at all to solve whole classes of DE's. You don't want to be thinking, 'really? That's the integral of tan(x), can that be possibly be right?' When you're supposed to be following a method.
Look at all the problems you ever solved in the rational functions and think, okay so what if this was a problem stated about the relationships between this function and its derivatives, what would that look like? If you studied a function and its derivatives and integrals in any prior units, get ready for those same things to reappear as a totally different-looking kind of problem and so the more you can get used to looking at functions and the whole family of curves related to those functions, looking for family resemblances, so to speak, by differentiation and integration and paying attention to what you see, the more ready you will be to absorb the methods for solving DE's and be in a position to enjoy that unit, rather than sweating on it.
Look into the Logistic Equation, be expecting that one. Also, one last plug for a really useful theorem most courses in integral calculus of a single variable seem to miss out on: Newton's Inverse Function Integration method is truly excellent. Take some nice function like y=sinx which maps x on to y, this has an inverse function, x=arcsiny which maps y on to x.
Newton's integral theorem says that if you know the integral of sinx , then you also know all you need to also integrate arcsiny.
My 2c.