Unformatted text preview: 5t e.g. y (t) = C1 e 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 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. 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. 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...
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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 UBC.
 Spring '13
 EricCytrynbaum
 Differential Equations, Equations, Complex Numbers

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