527_06exam1

527_06exam1 - Methods of Applied Mathematics— Math 527:01...

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Unformatted text preview: Methods of Applied Mathematics— Math 527:01 Exam 1 1. Find the power series expansion (Taylor series) centered at x = 1 for 1 3- x and give its radius of convergence. 2. (a) Find the Laplace transform of f ( t ) = braceleftbigg , if ≤ t < 1; e 2 t , if t ≥ 1 . (b) Find the inverse transform of 1 ( s- 3)( s 2 + 1) in the form of a single integral. (c) Use partial fractions, without computing the numerical values of the constants, to ex- press the inverse transform of 1 ( s + 2) 2 ( s 2 + 1) as a sum of functions. (The solution should have the form c 1 u 1 ( t ) + ··· + c k u k ( t ) , where the c 1 , . . . , c k are left undeter- mined.) 3. Use Laplace transforms to solve the initial value problem: y ′′ ( t ) + 2 y ′ ( t ) + 3 y ( t ) = δ ( t- 2) . y (0) = 1 , y ′ (0) =- 1 . 4. Consider the equation x 2 y ′′ + 2 x cos( x ) y ′ + 5 y = 0 . (a) x = 0 is a regular singular point. Explain why....
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527_06exam1 - Methods of Applied Mathematics— Math 527:01...

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