1 / 8

Annihilator Method

Annihilator Method. Jared Rosa Math 480 Spring 2013. Some Notations & Definitions. L(y) is the differential operator on y. E.g. y”’ - 3y” + 2y = x ; can be rewritten as L(y) = x. P c (r) is the characteristic polynomial of an ordinary differential equation.

amy
Download Presentation

Annihilator Method

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Annihilator Method Jared Rosa Math 480 Spring 2013

  2. Some Notations & Definitions • L(y) is the differential operator on y. • E.g. y”’ - 3y” + 2y = x ; can be rewritten as L(y) = x. • Pc(r) is the characteristic polynomial of an ordinary differential equation. • E.g. Pc(r) = r3 – 3r2 + 2 , is the characteristic polynomial of the ode y”’ - 3y” + 2y.

  3. So what is the Annihilator Method? • The Annihilator method is a faster method than the variation of constants used to solve non-homogenous differential equations with constant coefficients.

  4. How does the method work? • The annihilator method uses a second differential operator denoted M, where the solution of the ode M(b) = O is the right hand side of the first differential operator L(y). Then using the Pc(r) of M(L(y)) we can determine the solutions of L(y). • E.G. • L(y) = b(x) & M(b) = O • So we have • M(L(y)) = M(b) = O • Hence, M annihilates L.

  5. When Can I use this Method? • The annihilator method only has two simple requirement: • The differential operator that you want to annihilate has constant coefficients. E.g. L(y) = ay” + by’ + cy , where a, b, and c are constants • There exists a second differential operator M, such that if L(y) = f(x), then M(f) = O. Or said another way, the solution to the homogenous equation M(y) is f(x).

  6. List of Annihilators Function Pc(r) of an annihilator • r – a • (r-a)k+1 • r2 + a2 • (r2 + a2)k+1 • eax • xkeax • sin(ax), cos(ax) (a real) • xksin(ax), xkcos(ax) (a real)

  7. The Annihilator Vs. VOC Annihilator Method Variation Of Constants Method • L(y) = y”’ + y” + y’ + y = 1 • y(0) = 0, y’(o) = 1, y”(0) = 0 • First set L(y) = 0 to get Pc(r): • Pc(r) = r3+ r2 + r + 1 = (r2 + 1)(r + 1) • So, y1 = cos(x), y2 = sin(x), y3 = e-x • and yp = U1y1 + U2y2 + U3y3, where • U1 = ½(cos(x) – sin(x)) • U2 = ½(sin(x) + cos(x)) • U3 = ½e-x • So yg = 1 + c1sin(x) + c2cos(x) + c3e-x • And y = = 1 – ½ e-x + ½ sin(x) – ½ cos(x) • L(y) = y”’ + y” + y’ + y = 1 • y(0) = 0, y’(o) = 1, y”(0) = 0 • M(y) = y’ = 0 • M(L(y)) = y(4) + y”’ + y” + y’ = 0 • Pc(r) = r4 + r3 + r2 +r • 0 = r4 + r3 +r2 + r = r(r3 + r2 + r + 1) • 0 = r(r2 + 1)(r + 1) • yg = a + be-x + csin(x) + dcos(x) • We assume a=1 since a has to be a solution of L(y) to get:y = 1 – ½ e-x + ½ sin(x) – ½ cos(x)

  8. Source • Coddington, Earl A. An introduction to ordinary differential equations. Dover Publications, inc. New York: 1961. pg 86 – 92.

More Related