15 undetermined coefficients

15 undetermined coefficients - 18.03 Lecture#15 Oct 13 2009...

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18.03 Lecture #15 Oct. 13, 2009: notes Announcement: there were several errors on Problem Set 5. In problem 1(c), the t s should be x s. In problem 2(b), (c), the y s should be x s. These have been fixed on the web site as of Tuesday afternoon. The syllabus topic for today is the method of undetermined coefficients . The idea is very simple: in order to find a particular solution to an inhomogeneous (constant coefficient) linear ODE p ( D ) y = b ( t ) (Inhomogeneous constant coefficient) you guess the general shape of the solution—a formula involving several undetermined coefficients then plug your guess into the differential equation, and solve for the undetermined coefficients. If your guess wasn’t correct, then you won’t be able to solve the equations for the undetermined coefficients. If you can solve them, that proves that your guess was correct, and hands you the solution. The notation I’ll use today has independent variable t (instead of x ) because it’s often helpful to think of examples about systems evolving in time. The only serious new mathematics here is the Exponential Shift Rule below. Mostly this is a computational technique that requires you to practice it. So I’ll just do a very few examples (because you should do a lot of examples; some good sources are the problems in the supplementary notes 2F-6 and for section 2.5 of EP). Here is the simplest rule and a way to write down the corresponding calculations. Polynomial input rule. Suppose p is a non-zero polynomial, and b is a polynomial in t of degree k . As long as p (0) negationslash = 0, the equation p ( D ) y = b has a particular solution that is a polynomial in t of degree k . Example of computation. Suppose we want to solve y ′′ + 3 y + 2 y = t 2 t + 1 . The characteristic polynomial p is p ( D ) = D 2 + 3 D + 2. The input is a polynomial of degree two, so we look for a particular solution that’s a polynomial of degree two: y p = At 2 + Bt + C.
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