Question

2. (a) Find the Laplace transform of f(t) = e^{2t}u(t - 1), where u(t - 1) denotes the Heaviside step function. (b) Find the inverse Laplace transform of F(s) = frac{e^{-pi s}}{s^2 + 4s + 8}. (c) Use Laplace transforms to solve the ordinary differential equation frac{d^2x}{dt^2} + 4frac{dx}{dt} + 8x = delta(t - pi), where delta(t - pi) is the Dirac delta function, with initial conditions x(0) = 1 quad x'(0) = -2. Hint: Your result from part (b) may be useful.

          2. (a) Find the Laplace transform of
f(t) = e^{2t}u(t - 1),
where u(t - 1) denotes the Heaviside step function.
(b) Find the inverse Laplace transform of
F(s) = frac{e^{-pi s}}{s^2 + 4s + 8}.
(c) Use Laplace transforms to solve the ordinary differential equation
frac{d^2x}{dt^2} + 4frac{dx}{dt} + 8x = delta(t - pi),
where delta(t - pi) is the Dirac delta function, with initial conditions
x(0) = 1 quad x'(0) = -2.
Hint: Your result from part (b) may be useful.

$2. (a) Find the Laplace transform of f(t) = e^2tu(t - 1), where u(t - 1) denotes the Heaviside step function. (b) Find the inverse Laplace transform of F(s) = frace^-pi ss^2 + 4s + 8. (c) Use Laplace transforms to solve the ordinary differential equation fracd^2xdt^2 + 4fracdxdt + 8x = delta(t - pi), where delta(t - pi) is the Dirac delta function, with initial conditions x(0) = 1 quad x'(0) = -2. Hint: Your result from part (b) may be useful.$

Added by F-Tima N.

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