lecture10-2

lecture10-2 - Section 4.2(then 4.1 2nd order DFQ with...

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Unformatted text preview: Section 4.2 (then 4.1) 2nd order DFQ with constant coefficients Feb 24, 2009 Characteristic Method Today’s Session Section 4.2, cont’d A Summary of This Session: (0) Yet another example of “variation of parameters” (1) Solving constant coefficient homogeneous DFQ’s (1st order, second order, 3rd order, etc); (2) Characteristic Equation with real or repeated roots; (3) Concepts of linearity and independence; (4) nonhomogeneous DFQ with particular and general solutions. Characteristic Method Example (a) Verify that y 1 =- e x is a solution of dy dx = e 2 x + (1 + 2 e x ) y + y 2 (b) Let y = y 1 + 1 v be a solution of the DFQ. Find a differential equation satisfied by v . (c) Solve the differential equation in part (b) and write down the solution for the original DFQ. Answers: (b) v ′ + v =- 1; (c) v =- 1 + Ce − x and y =- e x + 1 Ce- x − 1 . Characteristic Method Back to Constant Coefficients Second Order DFQs Example 1: (a) Solve 6 y ′′- y ′- 2 y = 0. (b) Solve the IVP: 6 y ′′- y ′- 2 y = 0 subject to y (0) = 4 and y ′ (0) = 5. (c) Find the general solution of the nonhomogeneous problem 6 y ′′- y ′- 2 y = x 2- x + 3 ....
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lecture10-2 - Section 4.2(then 4.1 2nd order DFQ with...

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