Lecture 3 Notes

# Eq with constant coeff section 31 one case where the

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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)= e-t 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)= e-t 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)= e-t 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)= e-t 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)= e-t 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)= e-t 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.

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