Question

Use Laplace transforms to solve the following initial value problems, which explore variations on a theme: (c) +4y0,01

Please show all work and write clear.

Answer 1

y + 4y = e^-8t u(t) Apply laplace transform on both sides, we get = > Sy(s) - y(0) + 4y(s) = 1/s + 8 = > sy(s) - 1 + 4 y(s) = 1/s + 8 = > y(s) [s + 4] = 1/s + 8 + 1 = > y(s) [s + 4] = 1 + s + 8/s + 8 = > y(s) [s + 4] = s + 9/s + 8 = > y(s) = (s + 9)/(s + 4)(s + 8) y(s) = 5/4/s + 4 - 1/4/s + 8 y(s) = 1.25/s + 4 - 0.25/s + 8 By inverse laplace transform, y(t) = (1.25e^-4t - 0.25e^-8t) u(t) y + 4y = e^-4t u(t) Apply laplace transform on both sides we get = > sy(s) - y(0) + 4 y(s) = 1/s + 4 = > sy(s) - 1 + 4 y(s) = 1/s + 4 = > y(s) [s + 4] = 1/s + 4 + 1 y(s) [s + 4] = s + 5/s + 4