Introduction to Differential Equations - Self Quiz Solutions 38 | MATH 220, Study notes of Differential Equations

Download Introduction to Differential Equations - Self Quiz Solutions 38 | MATH 220 and more Study notes Differential Equations in PDF only on Docsity! MATH 220 Self-quiz 38 1. Find the Laplace transform of the function: f(t) =  t if 0 < t < π cos t if π < t < 2π et if t > 2π Solution: We begin by writing: f(t) = (1− u(t− π))t+ (u(t− π)− u(t− 2π)) cos t+ u(t− 2π)et = t+ u(t− π)(cos t− t) + u(t− 2π)(et − cos t) By using the formula: L{u(t− a)g(t)} = easL{g(t+ a)} we get; L{f(t)} = L{t}+ e−πsL{cos(t+ π)− (t+ π)}+ e−2πsL{et+2π − cos(t+ 2π)} = 1 s2 + e−πs ( − s s2 + 1 − 1 s2 − π s ) + e−2πs ( e2π s− 1 − s s2 + 1 ) 2. Find the inverse Laplace transform of the function F (s) = e−3s s3 . Solution: First we find the inverse Laplace of the function 1 s3 : L−1 { 1 s3 } = L−1 { 1 2 · 2 s3 } = t2 2