This preview shows page 1. Sign up to view the full content.
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...
View
Full
Document
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

Click to edit the document details