APM2611 Assignment 3 (COMPLETE ANSWERS) 2024 - DUE 14 August 2024
APM2611 Assignment 3 (COMPLETE ANSWERS) 2024 - DUE 14 August 2024 Question 1 1. Find the radius and interval of convergence of the following series: ( i) ∞X n=1 100n n! (x + 7) n (ii) ∞X k=1 (−1) k 10k (x − 5) k 2. Rewrite the expression below as a single power series: ∞X n=2 n(n − 1)cn x n + 2 ∞X n=2 n(n − 1)cn x n−2 + ∞X n=1 ncn x n . Question 2 1. Verify by direct substitution that the given power series is a particular solution of the DE (x + 1)y 00+ y 0 = 0 ; y = ∞X n=1 (−1) n+1 n x n . 2. Use the power series method to solve the initialvalue problem (x + 1)y 00 − (2 − x)y 0 + y = 0, y(0) = 2, y 0 (0) = −1; where c0 and c1 are given by the initial conditions. 16 APM2611/101/0/2024 Question 3 Calculate the Laplace transform of the following function from first principles: 1. f (t) = sin t if 0 ≤ t < π 0 if t ≥ π 2. f (t) = e −t sin t 3. Use Theorem 7.1 to find L{f (t)} (i) f (t) = −4t 2 + 16t + 9 (ii) f (t) = 4t 2 − 5 sin 3t (iii) f (t) = (e t − e −t ) 2 ### Questio
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apm2611 assignment 3 complete answers 2024 due