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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 < 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 < 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.
 Spring '13
 EricCytrynbaum
 Differential Equations, Equations, Complex Numbers

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