This preview shows page 1. Sign up to view the full content.
Unformatted text preview: 1 t r2 t e e Independence and the Wronskian (Section 3.2)
• So for case i (distinct roots), can we form a general solution from y1 (t) = er1 t y2 (t) = er2 t ? and • Must check the Wronskian: W (e r1 t r2 t ,e )(t) = e r1 t r2 t r1 t r 2 t − r1 e e r1 t r2 t = 0 r2 e = (r1 − r2 )e e Independence and the Wronskian (Section 3.2)
• So for case i (distinct roots), can we form a general solution from y1 (t) = er1 t y2 (t) = er2 t ? and • Must check the Wronskian: W (e r1 t r2 t )(t) = e r1 t r2 t r1 t r 2 t − r1 e e r1 t r2 t ,e = 0 r2 e = (r1 − r2 )e
So yes! y (t) = C1 e r1 t + C2 e r2 t e is the general solution. 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). 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). 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). 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). 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...
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 UBC.
 Spring '13
 EricCytrynbaum
 Differential Equations, Equations, Complex Numbers

Click to edit the document details