notes 4-4

# notes 4-4 - Diﬀerentiate y P and impose a condition on...

This preview shows pages 1–3. Sign up to view the full content.

SECTION 4.4 PARTICULAR SOLUTIONS BY VARIATION OF PARAMETERS We’re going to do the general third order case, then indicate how it generalizes for higher orders. So suppose we have a third order linear ordinary diﬀerential equation L [ y ] = y 000 + p ( t ) y 00 + q ( t ) y 0 + r ( t ) y = g ( t ) and suppose also that we have found a fundamental set of solutions y 1 ( t ) ,y 2 ( t ) ,y 3 ( t ) of L [ y ] = 0. We need a particular solution. A sum of constant multiples of y 1 ( t ) ,y 2 ( t ) ,y 3 ( t ) will not work, so we try the next best thing, namely, y P ( t ) = u 1 ( t ) y 1 ( t ) + u 2 ( t ) y 2 ( t ) + u 3 ( t ) y 3 ( t ) where u 1 ,u 2 ,u 3 are functions of t that we must ﬁnd. We’ll plug y P ( t ) into L [ y ] = g ( t ) and as we do so, we’ll impose conditions on the u i ’s to make our work easier. We have three unknown functions, so we can have three conditions.

This preview has intentionally blurred sections. Sign up to view the full version.

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

Unformatted text preview: Diﬀerentiate y P and impose a condition on the u i ’s to make the result nice. Do it again. Diﬀerentiate again and plug into L [ y ] = g ( t ). Notice that the coeﬃcient of y 000 is 1. Consolidate the equations for the derivatives of the u i ’s. Can we solve these equations for u 1 ,u 2 ,u 3 ? What would the equations look like in the ﬁfth order case? EXAMPLE. Find the general solution of y 000 + y = tan t,-π/ 2 < t < π/ 2 . HOMEWORK: SECTION 4.4...
View Full Document

{[ snackBarMessage ]}

### Page1 / 3

notes 4-4 - Diﬀerentiate y P and impose a condition on...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online