Lecture 4 Notes

# 2 y 9y 0 find the roots of the example consider the

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Unformatted text preview: ). Do these form a fundamental set of solutions? Calculate the Wronskian. W (cos(3t), sin(3t))(t) = Independence and the Wronskian (Section 3.2) y ￿￿ + 9y = 0. Find the roots of the • Example: Consider the equation characteristic equation (i.e. the r values). (A) r1 = 3, r2 = -3. (B) r1 = 3 (repeated root). (C) r1 = 3i, r2 = -3i. (D) r1 = 9, (repeated root). As we’ll see soon, this means that y1(t) = cos(3t) and y2(t)=sin(3t). Do these form a fundamental set of solutions? Calculate the Wronskian. W (cos(3t), sin(3t))(t) = (A) 0 (C) 3 (B) 1 (D) 2 cos(3t) sin(3t) Independence and the Wronskian (Section 3.2) y ￿￿ + 9y = 0. Find the roots of the • Example: Consider the equation characteristic equation (i.e. the r values). (A) r1 = 3, r2 = -3. (B) r1 = 3 (repeated root). (C) r1 = 3i, r2 = -3i. (D) r1 = 9, (repeated root). As we’ll see soon, this means that y1(t) = cos(3t) and y2(t)=sin(3t). Do these form a fundamental set of solutions? Calculate the Wronskian. W (cos(3t), sin(3t))(t) = (A) 0 (C) 3 (B) 1 (D) 2 cos(3t) sin(3t) Distinct roots - asymptotic behaviour (Section 3.1) • Three cases: Distinct roots - asymptotic behaviour (Section 3.1) • Three cases: (i) Both r values positive. e.g. y (t) = C1 e2t + C2 e5t Distinct roots - asymptotic behaviour (Section 3.1) • Three cases: (i) Both r values positive. e.g. y (t) = C1 e2t + C2 e5t Except for the zero solution y(t)=0, the limit lim y (t) ... t→∞ (A) ...is unbounded for all ICs. (B) ...is unbounded for most ICs but not for a few carefully chosen ones. (C) ...goes to zero for all ICs. Distinct roots - asymptotic behaviour (Section 3.1) • Three cases: (i) Both r values positive. e.g. y (t) = C1 e2t + C2 e5t Except for the zero solution y(t)=0, the limit lim y (t) ... t→∞ (A) ...is unbounded for all ICs. (B) ...is unbounded for most ICs but not for a few carefully chosen ones. (C) ...goes to zero for all ICs. Distinct roots - asymptotic behaviour (Section 3.1) • Three cases: (i) Both r values positive. e.g. y (t) = C1 e2t + C2 e...
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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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