4.3 Notes

4.3 Notes - Chapter 4 Higher-Order Diﬀerential Equations...

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Unformatted text preview: Chapter 4. Higher-Order Diﬀerential Equations §4.3 Homogeneous Linear Equations with Constant Coeﬃcients We study linear DEs in the form of dn y dn−1 y dy + an−1 n−1 + · · · + a1 + a0 y = 0 dxn dx dx where an , an−1 , · · · , a1 , a0 are constants. The equation an an λn + an−1 λn−1 + · · · + a1 λ + a0 = 0 is called the auxiliary equation of the DE. Ex: 3y ￿ + 5y = 0, 5y ￿￿ − 2y ￿ − y = 0, y ￿￿￿ + 5y ￿ − 8y = 0 1. Second order linear DE with constant coeﬃcients: The DE is ay ￿￿ + by ￿ + cy = 0. The auxiliary equation is aλ2 + bλ + c = 0. Three cases: • Two roots λ1 and λ2 are real and distinct (b2 − 4ac > 0). The general solution is y = c1 eλ1 x + c2 eλ2 x • Repeated real roots. i.e. λ1 = λ2 (b2 − 4ac = 0). The general solution is y = c1 eλ1 x + c2 xeλ2 x • Two complex roots, say λ1 = α + β i and λ2 = α − β i (b2 − 4ac < 0).The general solution is y = c1 eαx cos β x + c2 eαx sin β x 1 EX: Solve the DEs. (1) 2y ￿￿ − 5y ￿ − 3y = 0 (2) y ￿￿ − 10y ￿ + 25y = 0 (3) y ￿￿ + 4y ￿ + 7y = 0 (4) y ￿￿ + 16y = 0, y (0) = 2, y ￿ (0) = −2. (5) y ￿￿￿ + 3y ￿￿ − 4y = 0, (6) y (4) − 2y ￿￿ + y = 0. 2 ...
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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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