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Unformatted text preview: lucky. Homog. eq. with constant coeff. (Section 3.1)
• Which of the following functions are also solutions?
(A) y(t) = y1(t)2
(B) y(t) = y1(t)+y2(t)
(C) y(t) = y1(t) y2(t)
(D) y(t) = y1(t) / y2(t)
• In fact, the following are all solutions: C1y1(t), C2y2(t), C1y1(t)+C2y2(t).
• With ﬁrst order equations, the arbitrary constant appeared through an
integration step in our methods. With second order equations, not so lucky.
• Instead, ﬁnd two independent solutions, y1(t), y2(t), by whatever method. Homog. eq. with constant coeff. (Section 3.1)
• Which of the following functions are also solutions?
(A) y(t) = y1(t)2
(B) y(t) = y1(t)+y2(t)
(C) y(t) = y1(t) y2(t)
(D) y(t) = y1(t) / y2(t)
• In fact, the following are all solutions: C1y1(t), C2y2(t), C1y1(t)+C2y2(t).
• With ﬁrst order equations, the arbitrary constant appeared through an
integration step in our methods. With second order equations, not so lucky.
• Instead, ﬁnd two independent solutions, y1(t), y2(t), by whatever method.
• The general solution will be y(t) = C1y1(t) + C2y2(t). Homog. eq. with constant coeff. (Section 3.1)
• One case where the arbitrary constants DO appear as we calculate: y + y = 0 • More common would be that we ﬁnd solutions y(t) = 1 and y(t)= et and simply
write down Homog. eq. with constant coeff. (Section 3.1)
• One case where the arbitrary constants DO appear as we calculate: y + y = 0
y + y = C1
• More common would be that we ﬁnd solutions y(t) = 1 and y(t)= et and simply
write down Homog. eq. with constant coeff. (Section 3.1)
• One case where the arbitrary constants DO appear as we calculate: y + y = 0
y + y = C1
et y + et y = C1 et • More common would be that we ﬁnd solutions y(t) = 1 and y(t)= et and simply
write down Homog. eq. with constant coeff. (Section 3.1)
• One case where the arbitrary constants DO appear as we calculate: y + y = 0
y + y = C1
et y + et y = C1 et
(et y ) = C1 et • More common would be that we ﬁnd solutions y(t) = 1 and y(t)= et and simply
write down Homog. eq. with constant coeff. (Section 3.1)
• One case where the arbitrary constants DO appear as we calculate: y + y = 0
y + y = C1
et y + et y = C1 et
(et y ) = C1 et
et y = C1 et + C2
• More common would be that we ﬁnd solutions y(t) = 1 and y(t)= et and simply
write down Homog. eq. with constant coeff. (Section 3.1)
• One case where the arbitrary constants DO appear as we calculate: y + y = 0
y + y = C1
et y + et y = C1 et
(et y ) = C1 et
et y = C1 et + C2
y = C1 + C2 e−t
• More common would be that we ﬁnd solutions y(t) = 1 and y(t)= et and simply
write down Homog. eq. with constant coeff. (Section 3.1)
• One case where the arbitrary constants DO appear as we calculate: y + y = 0
y + y = C1
et...
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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

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