Problem #4: Consider y(t) = £¹ 3 2 3s-4 4 } (7) (a) Use partial fractions and find y(t). (b) Using the formula L{y'}(s) = s£{y}(s) − y(0), compute the Laplace transform of y'(1 be a function of s.

Suppose y satisfies the initial value problem ƒ y" (1) − 4y' (1) + 5y(t) = 8(t − 3) 1 y(0) = 0 and y'(0) = 1 Where is the delta function. Let Y(s) = L{ y(t)}. (a) By taking the Laplace transform of the ODE, show that how that Y(s) = (b) Determine an expression for y(t) by calculating £¯¹ {Y(s)} = e-3s +1 s² - 4s +5 Note: State each Laplace transform property as you use it. Refer to each property using its row number in the Table of Laplace Transforms provided. For example: "L{1} by [LT1]"

7. (20%) Using the Linearity of Laplace Transform (af (t)+bg(t))=ae {f(t)}+be{g(t)} (7a) and cosh at (e"+e) (7b) find (coshat) =] 9 coshat dt = (7c)