solut2 - Homework 2 Solutions 1 f x = 1 x x 2 x 6 = 1 x x 2...

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Unformatted text preview: Homework 2- Solutions. 1. f ( x ) = 1 x ( x 2- x- 6) = 1 x ( x + 2)( x- 3) = A x + B x + 2 + C x- 3 . Equations for the coefficients: x 2 ( A + B + C ) = 0 , x (- A- 3 B + 2 C ) = 0 , 1 (- 6 A ) = 1 . Solve: A =- 1 / 6, B = 1 / 10, C = 1 / 15. Have the indefinite integral, Z µ- 1 6 x + 1 10 1 x + 2 + 1 15 1 x- 3 ¶ dx =- 1 6 log | x | + 1 10 log | x + 2 | + 1 15 log | x- 3 | . Next, we evaluate the definite integral.- 1 6 log | x | fl fl fl fl 2 1 =- 1 6 (log2- log1) =- 1 6 log2 . 1 10 log | x + 2 | fl fl fl fl 2 1 = 1 10 (log4- log3) = 1 10 log(4 / 3) . 1 15 log | x- 3 | fl fl fl fl 2 1 = 1 15 (log1- log2) =- 1 15 log2 . All together,- 1 6 log2 + 1 10 log(4 / 3)- 1 15 log2 =- 7 30 log2 + 1 10 log(4 / 3) . 4 Section 3.2, Problem 32. Write down the right hand side in terms of the Heaviside function, f ( t ) = 12 H ( t- 4) . Calculate Laplace of f ( t ): L [ f ] = L [12 H ( t- 4)] = 12 L [ H ( t- 4)] = 12 1 s e- 4 s , 1 see Example 3.11. Take Laplace of both sides of the equation. Set L...
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solut2 - Homework 2 Solutions 1 f x = 1 x x 2 x 6 = 1 x x 2...

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