hw5soln-corrected

3 y e t 3 7 cos t sin t 116 116 the sum of

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: os (t ) + −β sin (t ) ; and therefore ￿2 ￿ ￿￿ ￿ ￿ ￿ ￿ D − 4D + 10 y = e −t 14α − 6β cos (t ) + 6α + 14β sin (t ) . 3 (3.3) Substitute this expression into (3.1). e −t Solve for α and β. ￿￿ ￿ ￿ ￿ ￿ 14α − 6β cos (t ) + 6α + 14β sin (t ) = e −t sin (t ) 14α − 6β = 0 6 α = 14 β ... ... ... 6 7 α = 14 · 116 3 = 116 and and . . . and 6 ￿ 6 6α + 14β = 1 ￿ β + 14β = 1 14 116 7β =1 7 β = 116 Substitute these values into (3.3). y =e −t ￿ 3 7 cos (t ) + sin (t ) 116 116 ￿ The sum of this solution with (3.2) is the general solution. ￿ ￿ ￿ ￿￿ ￿ ￿￿ ￿￿ 7 3 y = e −t cos (t ) + sin (t ) + e 2t C 1 cos 6t + C 2 sin 6t 116 116 4 4 The equation to solve is ￿3 ￿ D − 2D 2 − 4D + 8 y = 6xe 2x . (4.1) Solve the associated homogeneous equation. ￿ ￿ D 3 − 2D 2 − 4D + 8 y ∗ = 0 (D − 2)2 (D + 2) y ∗ = 0 y ∗ = (C 0 + C 1 x ) e 2x + C 2 e −2x (4.2) ￿ ￿ The obvious judicious guess, y = e 2x αx + β , won’t work since it’s already a homogeneous solution. So instead use ￿ ￿ y = e 2x α x 3 + β x 2 . (4.3) For convenience, substitute s = αx 3 + βx 2 . Then Combining these gives ￿ y = e 2x s ; ￿ ￿ D...
View Full Document

Ask a homework question - tutors are online