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Unformatted text preview: ¿¿ï¿¿ ï¿¿ rt we get the characteristic equation: ar + br + c = 0
2 â€¢ There are three cases.
i. Two distinct real roots: b2  4ac > 0. ( r1 â‰ r2 )
ii.A repeated real root: b2  4ac = 0.
iii.Two complex roots: b2  4ac < 0. y1 (t) = er1 t
â€¢ For case i, we get and y2 (t) = er2 t. â€¢ 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) = e2tâˆ’3 are two
â€¢ Example: Suppose
solutions to some equation. Can we solve ANY initial
y
condition . (0) = y0 , y ï¿¿ (0) = v0 with these two solutions?
. â€¢ Solve this system for C1, C2...
â€¢ Canâ€™t do it. Why? Independence and the Wronskian (Section 3.2)
y1 (t) = e2t+3 and y2 (t) = e2tâˆ’3 are two
â€¢ Example: Suppose
solutions to some equation. Can we solve ANY initial
y
condition . (0) = y0 , y ï¿¿ (0) = v0 with these two solutions?
. y (t) = C1 e2t+3 + C2 e2tâˆ’3 â€¢ Solve this system for C1, C2...
â€¢ Canâ€™t do it. Why? Independence and the Wronskian (Section 3.2)
y1 (t) = e2t+3 and y2 (t) = e2tâˆ’3 are two
â€¢ Example: Suppose
solutions to some equation. Can we solve ANY initial
y
condition . (0) = y0 , y ï¿¿ (0) = v0 with these two solutions?
. y (t) = C1 e2t+3 + C2 e2tâˆ’3
y (0) = C1 e3 + C2 eâˆ’3 = y0 â€¢ Solve this system for C1, C2...
â€¢ Canâ€™t do it. Why? Independence and the Wronskian (Section 3.2)
y1 (t) = e2t+3 and y2 (t) = e2tâˆ’3 are two
â€¢ Example: Suppose
solutions to some equation. Can we solve ANY initial
y
condition . (0) = y0 , y ï¿¿ (0) = v0 with these two solutions?
. y (t) = C1 e2t+3 + C2 e2tâˆ’3
y (0) = C1 e3 + C2 eâˆ’3 = y0 y ï¿¿ (0) = 2C1 e3 + 2C2 eâˆ’3 = v0
â€¢ Solve this system for C1, C2...
â€¢ Canâ€™t do it. Why? Independence and the Wronskian (Section 3.2)
y1 (t) = e2t+3 and y2 (t) = e2tâˆ’3 are two
â€¢ Example: Suppose
solutions to some equation. Can we solve ANY initial
y
condition . (0) = y0 , y ï¿¿ (0) = v0 with these two solutions?
. y (t) = C1 e2t+3 + C2 e2tâˆ’3
y (0) = C1 e3 + C2 eâˆ’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.
 Spring '13
 EricCytrynbaum
 Differential Equations, Equations, Complex Numbers

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