Lecture 3 Notes

Plug it in and check ac1 y1 bc1 y1 cc1 y1

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Unformatted text preview: y1(t) and y2(t). This means that ￿￿ ￿ ay1 + by1 + cy1 = 0 and ￿￿ ￿ ay2 + by2 + cy2 = 0 Homog. eq. with constant coeff. (Section 3.1) ay ￿￿ + by ￿ + cy = 0 • Suppose you already found a couple solutions, y1(t) and y2(t). This means that ￿￿ ￿ ay1 + by1 + cy1 = 0 and ￿￿ ￿ ay2 + by2 + cy2 = 0 • Notice that y(t) = C1y1(t) is also a solution. Plug it in and check: Homog. eq. with constant coeff. (Section 3.1) ay ￿￿ + by ￿ + cy = 0 • Suppose you already found a couple solutions, y1(t) and y2(t). This means that ￿￿ ￿ ay1 + by1 + cy1 = 0 ￿￿ ￿ ay2 + by2 + cy2 = 0 and • Notice that y(t) = C1y1(t) is also a solution. Plug it in and check: a(C1 y1 ) + b(C1 y1 ) + c(C1 y1 ) ￿￿ ￿ Homog. eq. with constant coeff. (Section 3.1) ay ￿￿ + by ￿ + cy = 0 • Suppose you already found a couple solutions, y1(t) and y2(t). This means that ￿￿ ￿ ay1 + by1 + cy1 = 0 ￿￿ ￿ ay2 + by2 + cy2 = 0 and • Notice that y(t) = C1y1(t) is also a solution. Plug it in and check: a(C1 y1 ) + b(C1 y1 ) + c(C1 y1 ) ￿￿ ￿ = aC1 (y1 ) + bC1 (y1 ) + cC1 (y1 ) ￿￿ ￿ Homog. eq. with constant coeff. (Section 3.1) ay ￿￿ + by ￿ + cy = 0 • Suppose you already found a couple solutions, y1(t) and y2(t). This means that ￿￿ ￿ ay1 + by1 + cy1 = 0 ￿￿ ￿ ay2 + by2 + cy2 = 0 and • Notice that y(t) = C1y1(t) is also a solution. Plug it in and check: a(C1 y1 ) + b(C1 y1 ) + c(C1 y1 ) ￿￿ ￿ = aC1 (y1 ) + bC1 (y1 ) + cC1 (y1 ) ￿￿ ￿ ￿￿ ￿ = C1 (ay1 + by1 + cy1 ) Homog. eq. with constant coeff. (Section 3.1) ay ￿￿ + by ￿ + cy = 0 • Suppose you already found a couple solutions, y1(t) and y2(t). This means that ￿￿ ￿ ay1 + by1 + cy1 = 0 ￿￿ ￿ ay2 + by2 + cy2 = 0 and • Notice that y(t) = C1y1(t) is also a solution. Plug it in and check: a(C1 y1 ) + b(C1 y1 ) + c(C1 y1 ) ￿￿ ￿ = aC1 (y1 ) + bC1 (y1 ) + cC1 (y1 ) ￿￿ ￿ ￿￿ ￿ = C1 (ay1 + by1 + cy1 ) = 0 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) 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) 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). 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...
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