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Exercise Set 3

# Exercise Set 3 - Exercise Set 3 Math 4027 Due 1 Verify that...

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Unformatted text preview: Exercise Set 3 Math 4027 Due: March 15, 2005 1. Verify that the function ϕ 1 ( x ) is a solution of the given differential equation, and find a second linearly independent solution ϕ 2 ( x ) on the interval indicated. (a) y 00- 2 x- 2 y = 0 ϕ 1 ( x ) = x 2 < x < ∞ (b) y 00- 4 xy + (4 x 2- 2) y = 0 ϕ 1 ( x ) = e x 2-∞ < x < ∞ (c) (1- x 2 ) y 00- 2 xy + 2 y = 0 ϕ 1 ( x ) = x < x < 1 (a) I Solution. ϕ 00 1 ( x )- 2 x- 2 ϕ 1 ( x ) = 2- 2 = 0 so ϕ 1 ( x ) is a solution to y 00- 2 x- 2 y = 0 . If we define ϕ 2 = ϕ 1 u , then ϕ 2 ( x ) = x 2 u ϕ 2 ( x ) = 2 xu + x 2 u ϕ 00 2 ( x ) = 2 u + 4 xu + x 2 u 00 . If ϕ 2 is to be a solution of the differential equation y 00- 2 x- 2 y = 0, then we must have 0 = ϕ 00 2 ( x )- 2 x- 2 ϕ 2 ( x ) = 2 u- 2 u + 4 xu + x 2 u 00 . This means that v = u satisfies the first order linear equation x 2 v + 4 xv = 0. Writing this in standard form gives v + 4 x- 1 v = 0 which means that there is an integrating factor I ( x ) = x 4 . That is ( x 4 v ) = 0 so that x 4 v = C where C is a constant. Then u = v = Cx- 4 so that u =- C 3 x- 3 + B for B a constant. Choosing C =- 3 and B = 0 gives one choice for u , namely u = x- 3 which gives a second solution ϕ 2 ( x ) = x 2 u = 1 x . J (b) I Solution. Computing the derivatives of ϕ 1 ( x ) gives ϕ 1 ( x ) = e x 2 ϕ 1 ( x ) = 2 xe x 2 ϕ 00 1 ( x ) = (2 + 4 x 2 ) e x 2 , which gives ϕ 00 1 ( x )- 4 xϕ 2...
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Exercise Set 3 - Exercise Set 3 Math 4027 Due 1 Verify that...

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