a) Use the Laplace transform on the second order ODE, y (t) + ay'(t) + by(t)...

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a) Use the Laplace transform on the second order ODE, y" (t) + ay'(t) + by(t) = q(t), y(0) = u, y'(0) = v, a, b e R, > to show that u(s + a) + v + L(q(t)) Y(s) := L(y(t)) = a? (s + )² + b – 4 b) Show that Y (s) in the previous point can be written as, s+ L(q(t)) 1 Z(s) = u- (s + )² + (b – 4) (s + )² + (b- (s + )² + (b – $)' and solve the problem when q(t) = 0. a) Use the Laplace transform on the second order ODE, y" (t) + ay'(t) + by(t) = q(t), y(0) = u, y'(0) = v, a, b e R, > to show that u(s + a) + v + L(q(t)) Y(s) := L(y(t)) = a? (s + )² + b – 4 b) Show that Y (s) in the previous point can be written as, s+ L(q(t)) 1 Z(s) = u- (s + )² + (b – 4) (s + )² + (b- (s + )² + (b – $)' and solve the problem when q(t) = 0.

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