notes 4-4 - Dierentiate y P and impose a condition on the u...

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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.
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Unformatted text preview: Dierentiate y P and impose a condition on the u i s to make the result nice. Do it again. Dierentiate again and plug into L [ y ] = g ( t ). Notice that the coecient 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 fth order case? EXAMPLE. Find the general solution of y 000 + y = tan t,-/ 2 < t < / 2 . HOMEWORK: SECTION 4.4...
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notes 4-4 - Dierentiate y P and impose a condition on the u...

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