Math 331 Homework 10: Calculating Laplace Transforms using Different Methods, Assignments of Mathematics

Download Math 331 Homework 10: Calculating Laplace Transforms using Different Methods and more Assignments Mathematics in PDF only on Docsity! Math 331 – Homework #10 (1) Let b and c be real constants. Calculate L ( ebt cosh(ct) ) using the following two different methods. Show that the two methods give the same answer. Method #1. Using the formula cosh(ct) = 1 2 ect+1 2 e−ct, write ebt cosh(ct) as 1 2 ×(sum of two exponentials) and use the linearity of L and the formula L(eat) = 1/(s− a), where a is a real constant. Method #2. Use the formula for L(cosh(ct)) derived in class together with the first part of Theorem 6.3.2. (2) §6.1, #15. Calculate L(teat) using the first part of Theorem 6.3.2. (3) §6.2, #10. Calculate L−1 ( 2s − 3 s2 + 2s + 10 ) by completing the square in the denominator and using the second part of Theorem 6.3.2. (4) §6.3, #13, #14, #15, #17 with each exponential term replaced by 1. These are the following inverse Laplace transforms. (a) Use the second part of Theorem 6.3.2 to calculate L−1 ( 3! (s − 2)4 ) . (b) Factor the denominator and use partial fractions to calculate L−1 ( 1 s2 + s − 2 ) . (c) Complete the square in the denominator and use the second part of Theorem 6.3.2 to calculate L−1 ( 2(s − 1) s2 − 2s + 2 ) . (d) Either complete the square in the denominator and use the second part of Theorem 6.3.2 or factor the denominator and use partial fractions to calculate L−1 ( s − 2 s2 − 4s + 3 ) . 1