section%204.4

section%204.4 - Chapter 4. Higher-Order Differential...

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Unformatted text preview: Chapter 4. Higher-Order Differential Equations §4.4 Undetermined Coefficients–Superposition Approach We study linear nonhomogeneous DEs in the form of dn y dn−1 y dy + an−1 n−1 + · · · + a1 + a0 y = g (x) dxn dx dx where an , an−1 , · · · , a1 , a0 are constants. The general solution of the DE is y = yc + yp an where yc is the general solution of the corresponding homogeneous DE. EX: Find a particular solution and then the general solution. (1) y ￿￿ − 2y ￿ − 3y = x + 1 (2) y ￿￿ − 2y ￿ + y = 2 sin x Trial Particular Solutions g (x) c 5x + 7 3x2 − 3x + 1 sin 2x(cos 2x) e3x 6xe3x e3x sin 2x xex cos x yp A Ax + B Ax2 + Bx + C A sin 2x + B cos 2x Ae3x (Ax + B )e3x Ae3x sin 2x + Be3x cos 2x (Ax + B )ex sin x + (Cx + D)ex cos x EX: Find a particular solution and then the general solution. y ￿￿ + y ￿ = x + 5 − e2x y ￿￿ − 2y ￿ + y = ex 1 ...
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This note was uploaded on 01/05/2011 for the course MATH 3310 taught by Professor Dr.du during the Fall '08 term at Kennesaw.

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