Self-Quiz 40 Solutions - Introduction to Differential Equations | MATH 220, Study notes of Differential Equations

Download Self-Quiz 40 Solutions - Introduction to Differential Equations | MATH 220 and more Study notes Differential Equations in PDF only on Docsity! MATH 220 Self-quiz 40 1. Find the Laplace transform of the function: f(t) = { t if 0 < t < 2 t2 if 2 < t Solution: We begin by writing: f(t) = t(1− u(t− 2)) + t2u(t− 2) where, as usual: u(t) = { 1 t > 0 0 t < 0 This can be rewritten as: f(t) = t + u(t− 2)(t2 − t) We can now compute the Laplace transform: L{f(t)} = 1 s2 + e−2sL{(t + 2)2 − (t + 2)} Thus: L{f(t)} = 1 s2 + e−2sL{t2 + 3t + 2} = 1 s2 + e−2s ( 2 s3 + 3 s2 + 2 s ) 2. Use the power series method in order to solve the initial value problem: y′ = 3y y(0) = 1 Solution: We are going to look for a solution in the form: y(x) = +∞∑ n=0 anx n We have that: y′(x) = +∞∑ n=1 nanx n−1 and by substituting into our equation we get: +∞∑ n=1 nanx n−1 = 3 +∞∑ n=0 anx n We reindex the summation on the left: +∞∑ n=0 (n + 1)an+1xn = 3 +∞∑ n=0 anx n and comparison of coefficients yields: (n + 1)an+1 = 3an Also, the initial condition y(0) = 1 can be interpreted as ao = 1. By multiplying the equations: 2