Lecture 4 Notes

Is unbounded for most ics but not for a few carefully

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Unformatted text preview: ii) The r values have opposite sign. e.g. y (t) = C1 e −2t t→−∞ (A) ...is unbounded for all ICs. (ii) Both r values negative. −2t Except for the zero solution y(t)=0, the limit lim y (t) ... + C2 e 5t (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 e.g. y (t) = C1 e + C2 e −5t (iii) The r values have opposite sign. e.g. y (t) = C1 e −2t t→−∞ (A) ...is unbounded for all ICs. (ii) Both r values negative. −2t Except for the zero solution y(t)=0, the limit lim y (t) ... + C2 e 5t (B) ...is unbounded for most ICs but not for a few carefully chosen ones. (C) ...goes to zero for all ICs. Complex roots (Section 3.3) • Complex number review (Euler’s formula) • Complex roots of the characteristic equation • From complex solutions to real solutions Complex number review • We deﬁne a new number: i= √ −1 Complex number review • We deﬁne a new number: i= √ −1 • Before, we would get stuck solving any equation that required squarerooting a negative number. No longer. Complex number review • We deﬁne a new number: i= √ −1 • Before, we would get stuck solving any equation that required squarerooting a negative number. No longer. • e.g. The solutions to x2 − 4x + 5 = 0 are x=2+i and x=2−i Complex number review • We deﬁne a new number: i= √ −1 • Before, we would get stuck solving any equation that required squarerooting a negative number. No longer. • e.g. The solutions to x2 − 4x + 5 = 0 are x=2+i and x=2−i ax2 + bx + c = 0 , when b2 - 4ac &lt; 0, the solutions • For any equation, have the form x = α ± β i where α and β are both real numbers. Complex number review • We deﬁne a new number: i= √ −1 • Before, we would get stuck solving any equation that required squarerooting a negative number. No longer. • e.g. The solutions to x2 − 4x + 5 = 0 are x=2+i and x=2−i ax2 + bx + c = 0 , when b2 - 4ac &lt; 0, the solutions • For any equation, have the form x = α ± β i where α and β are both real numbers. • For α+βi, we call α the real part and β the imaginary part. Complex number review (a + bi) + (c + di) = a + c + (b + d)i Complex number review • Adding two complex numbers: (a + bi) + (c + di) = a + c + (b + d)i Complex number review • Adding two complex numbers: (a + bi) + (...
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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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