Question

Consider the following initial value problem, representing the response of a damped oscillator subject to the discontinuous applied force f(t): y" +2y +10y = f(t), y(0) = 6, 7(0) = -3, f(t) = (1 3<t<4, 10 otherwise. {o In the following parts, use h(t -c) for the Heaviside function he(t) when necessary. a. First, compute the Laplace transform of f(t). L{f(t)}(s) = b. Next, take the Laplace transform of the left-hand-side of the differential equation, set it equal to your answer from Part a. and solve for L{y(t)}. L{y(t)}(s) = c. We will need to take the inverse Laplace transform in order to find y(t). To do so, let's first rewrite L{y(t)} as L{y(t)}(s) = 1 -e 10 ***(4-F(s) Hoe (4 – F(s)) + 6F(a) – 3G(0) where F(S) = and G(s) = d. Part с. indicates we will need L '{F(s)} and L '{G(s)}. Let's go ahead and find those now. L '{F(s)}(t) = and < '{G(s)}(t) = e. Use your answer in Part d. to compute '{e 38F(s)} and L '{e 48 F(s)}. L'{e 38F(s)}(t) = and < '{e 48F(s)}(t) = f. Finally, combine all the previous steps to write down y(t). g(t) =

Answer 1