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Unformatted text preview: 3.3: Homogeneous Equations with Constant Coefficients Remark Assume that y H x L = e r x is a solution to the n th order, linear, homogeneous differential equation a n y H n L + + a y = 0. Then, since d i dx i e r x = r i e rx , for 1 i n , we have a n r n e r x + a n- 1 r n- 1 e r x + + a e r x = if andonly if a n r n + a n- 1 r n- 1 + + a = 0. Definition The characteristic equation for the n th order, linear, homogeneous differential equation a n y H n L + + a y = 0. is given by a n r n + a n- 1 r n- 1 + + a = 0. Theorem - Solutions to the n th Order, Linear, Homogeneous DE with Constant Coefficients: Distinct and Real If the roots r 1 , ..., r n of the characteristic equation are real and distinct, Then y H x L = c 1 e r 1 x + c 2 e r 2 x + + c n e r n x is a general solution to a n y H n L + + a y = 0. Example Solve the IVP y H 3 L + 2 y ''- 5 y '- 6 y = 0, y H L = 1, y ' H L = 2, y '' H L = 3. Mathematica Example Solve the IVP y H 3 L + 2 y ''- 5 y '- 6 y = 0, y H L = 1, y ' H L = 2, y '' H L = 3 using Mathematica . First, let's factor the characteristic equation... Factor @ r^3 + 2 r^2- 5 r- 6 D H- 2 + r L H 1 + r L H 3 + r L Or we could just solve......
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This note was uploaded on 01/13/2012 for the course MATH 306 taught by Professor Keithemmert during the Spring '11 term at Tarleton.
- Spring '11