Lecture 4 Notes

Can we solve any initial y condition 0 y0 y 0 v0

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Unformatted text preview: = y0 y (0) = 2C1 e3 + 2C2 e−3 = v0 • Solve this system for C1, C2... Independence and the Wronskian (Section 3.2) y1 (t) = e2t+3 and y2 (t) = e2t−3 are two • Example: Suppose solutions to some equation. Can we solve ANY initial y condition . (0) = y0 , y ￿ (0) = v0 with these two solutions? . y (t) = C1 e2t+3 + C2 e2t−3 y (0) = C1 e3 + C2 e−3 = y0 y ￿ (0) = 2C1 e3 + 2C2 e−3 = v0 • Solve this system for C1, C2... • Can’t do it. Why? Independence and the Wronskian (Section 3.2) y1 (t) = e2t+3 and y2 (t) = e2t−3 are two • Example: Suppose solutions to some equation. Can we solve ANY initial y condition . (0) = y0 , y ￿ (0) = v0 with these two solutions? . y (t) = C1 e2t+3 + C2 e2t−3 y (0) = C1 e3 + C2 e−3 = y0 y ￿ (0) = 2C1 e3 + 2C2 e−3 = v0 • Solve this system for C1, C2... • Can’t do it. Why? ￿ e3 3 e ￿￿ ￿ ￿ ￿ e−3 C1 y0 = −3 e C2 v0 Independence and the Wronskian (Section 3.2) y1 (t) = e2t+3 and y2 (t) = e2t−3 are two • Example: Suppose solutions to some equation. Can we solve ANY initial y condition . (0) = y0 , y ￿ (0) = v0 with these two solutions? . y (t) = C1 e2t+3 + C2 e2t−3 y (0) = C1 e3 + C2 e−3 = y0 y ￿ (0) = 2C1 e3 + 2C2 e−3 = v0 • Solve this system for C1, C2... • Can’t do it. Why? ￿ ￿￿ ￿ ￿ ￿ e3 e−3 C1 y0 = 3 −3 ee C2 v0 ￿3 ￿ e e−3 det 3 =0 −3 ee Independence and the Wronskian (Section 3.2) • For any two solutions to some linear ODE, to ensure that we have a general solution, we need to check that ￿ ￿ y1 (0) y2 (0) ￿ ￿ det ￿ = y1 (0)y2 (0) − y1 (0)y2 (0) ￿= 0 ￿ y1 (0) y2 (0) Independence and the Wronskian (Section 3.2) • For any two solutions to some linear ODE, to ensure that we have a general solution, we need to check that ￿ ￿ y1 (0) y2 (0) ￿ ￿ det ￿ = y1 (0)y2 (0) − y1 (0)y2 (0) ￿= 0 ￿ y1 (0) y2 (0) Independence and the Wronskian (Section 3.2) • For any two solutions to some linear ODE, to ensure that we have a general solution, we need to check that ￿ ￿ y1 (0) y2 (0) ￿ ￿ det ￿ = y1 (0)y2 (0) − y1 (0)y2 (0) ￿= 0 ￿ y1 (0) y2 (0) • For ICs other t...
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