hw_3_24_09

hw_3_24_09 - MA 221 Homework Solutions Due date: March 24,...

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MA 221 Homework Solutions Due date : March 24 , 2009 8 . 3p . 449 # 1 , 3 ̄ , 5 ̄ , 7 , 11 , 1 ̄ 2 ̄ , 1 ̄ 5 ̄ , 1 ̄ 7 ̄ , 1 ̄ 9 ̄ , 2 ̄ 1 ̄ , 25 2 ̄ 7 (Underlined problems are to be handed in) In problems 1, 3, 5 and 7 Determine all the singular points of the given differential equations. 1.) x 1 y ′′ x 2 y 3 y 0 Dividing the entire equation by x 1 yields y ′′ x 2 x 1 y 3 x 1 y 0 We then see: P y x 2 x 1 Q y 3 x 1 These are rational functions and so they are analytical everywhere except, perhaps, at zeros of their denominators. Solving x 1 0wefindthat x 1 which is at a point of infinite discontinuity for both functions. Consequently, x 1 is the only singular point of the given equation. 3 ̄ .) 2 2 y ′′ 2 y sin y 0 Writing the equation in standard form yields y ′′ 2 2 2 y sin 2 2 y 0 and P 2 2 2 Q sin 2 2 The singularities are therefore at  2 . Find at least the first four nonzero terms in a power series expansion about x 0 for a general solution to the given differential equation. 5 ̄ .) t 2 t 2 x ′′ t 1 x t 2 x 0 x ′′ t 1 t 2 t 2 x t 2 t 2 t 2 x 0 p t t 1 t 2 t 2 t 1 t 1  t 2 q t t 2 t 2 t 2 t 2 t 1  t 2 The point t 1 is a removable singularity for p t since, for t ≠− 1, we can cancel t 1 term in the numerator and denominator, and so p t becomes analytic at t 1ifweset 1
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p 1 : lim t 1 p t lim t 1 1 t 2 1 3 At the point t 2, p t has infinite discontinuity. Thus p t is analytic everywhere except t 2. Similarly, q t is analytic everywhere except t 1. Therefore, the given equation has two singular points, t 1and t 2. 7.) sin x y ′′ cos x y 0 Putting the equation in standard form we get: y ′′ cos x sin x y 0 Hence: p x 0 q x cos x sin x cot x Since the cotangent function is at integer multiples of , we see that q x is not defined and , therefore not analytical at n . Hence the differential equation is singular only at the points n ,where n is an integer. In problems 11, 12, 15 and 17 find at least the first four non zero terms in a power series expansion about x 0 for a general solution to the given differential equation. 11.)
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This note was uploaded on 09/22/2009 for the course PEP 112 taught by Professor Whittaker during the Fall '07 term at Stevens.

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hw_3_24_09 - MA 221 Homework Solutions Due date: March 24,...

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