Two distinct real roots b2 4ac 0 r1 r2 iia repeated

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Unformatted text preview: ￿ ￿ rt we get the characteristic equation: ar + br + c = 0 2 • There are three cases. i. Two distinct real roots: b2 - 4ac > 0. ( r1 ≠ r2 ) ii.A repeated real root: b2 - 4ac = 0. iii.Two complex roots: b2 - 4ac < 0. y1 (t) = er1 t • For case i, we get and y2 (t) = er2 t. • Do our two solutions cover all possible ICs? That is, can we use them to form a general solution? Independence and the Wronskian (Section 3.2) y1 (t) = e2t+3 and y2 (t) = e2t−3 are two • Example: Suppose solutions to some equation. Can we solve ANY initial y condition . (0) = y0 , y ￿ (0) = v0 with these two solutions? . • Solve this system for C1, C2... • Can’t do it. Why? Independence and the Wronskian (Section 3.2) y1 (t) = e2t+3 and y2 (t) = e2t−3 are two • Example: Suppose solutions to some equation. Can we solve ANY initial y condition . (0) = y0 , y ￿ (0) = v0 with these two solutions? . y (t) = C1 e2t+3 + C2 e2t−3 • Solve this system for C1, C2... • Can’t do it. Why? Independence and the Wronskian (Section 3.2) y1 (t) = e2t+3 and y2 (t) = e2t−3 are two • Example: Suppose solutions to some equation. Can we solve ANY initial y condition . (0) = y0 , y ￿ (0) = v0 with these two solutions? . y (t) = C1 e2t+3 + C2 e2t−3 y (0) = C1 e3 + C2 e−3 = y0 • Solve this system for C1, C2... • Can’t do it. Why? Independence and the Wronskian (Section 3.2) y1 (t) = e2t+3 and y2 (t) = e2t−3 are two • Example: Suppose solutions to some equation. Can we solve ANY initial y condition . (0) = y0 , y ￿ (0) = v0 with these two solutions? . y (t) = C1 e2t+3 + C2 e2t−3 y (0) = C1 e3 + C2 e−3 = y0 y ￿ (0) = 2C1 e3 + 2C2 e−3 = v0 • Solve this system for C1, C2... • Can’t do it. Why? Independence and the Wronskian (Section 3.2) y1 (t) = e2t+3 and y2 (t) = e2t−3 are two • Example: Suppose solutions to some equation. Can we solve ANY initial y condition . (0) = y0 , y ￿ (0) = v0 with these two solutions? . y (t) = C1 e2t+3 + C2 e2t−3 y (0) = C1 e3 + C2 e−3...
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This note was uploaded on 02/12/2014 for the course MATH 256 taught by Professor Ericcytrynbaum during the Spring '13 term at The University of British Columbia.

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