Lecture 4 Notes

Y t c1 e r1 t c2 e r2 t e is the general solution

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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...
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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 UBC.

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