Solve the equation using the Laplace transformation and show

Solve the equation using the Laplace transformation and show the work please

Solve the equation using the Laplace transformation and show the work 4x + x + 16x = e^-t where x(0) = 2 and x(0) = 0

Solution

L[4x\'\' + x\' + 16x] = L[e^-t]

L[e^-t] = 1/(s + 1)

L(x) = Y(s)

L(x\') = s*L(x) - x(0) = s*Y(s) - 2

L(x\'\') = s^2*L(x) - s*x(0) - x\'(0)= s^2*Y(s) - 2s - 0

L(x\'\') = s^2*Y(s) - 2s

4s^2*Y(s) - 8s + s*Y(s) - 2 + 16*Y(s) = 1/(s + 1)

(4s^2 + s + 16)*Y(s) - (8s + 2) = 1/(s + 1)

(4s^2 + s + 16)*Y(s) = 1/(s + 1) + (8s + 2)

Y(s) = [1/(s + 1)*(4s^2 + s + 16)] + [(8s + 2)/(4s^2 + s + 16)]

Y(s) =