Question

Use the Laplace Transform method to solve the IVP y" - 8y + 16y = t4 y(0) = 1,5(0) - 4. Show all your work Note: A partial fraction decomposition will not be needed here if you carefully solve for Y(s) = {y}(s), by first moving the expression of the form -as - b with a and b positive integers to the right hand side and then dividing both sides of the equation by the coefficient of Y() which will be of the form s2 - 2ms+ma b (8 - m)? At this time, check your work by verifying that Y () is of the form (8 - )* (8 m) and proceed to find the solution to the IVP by computing the inverse Laplace transform of Y(s). a

Answer 1

The given differential equation is y" - 8y' +167 teht 9() = 1, Y'60)=4 Taking Lab luce transformation both sides we got tfyl} – BL{y&} +16Lsy) = +84) = ŠL{y}-87(0) –Y'(o) for [$117)- you] 2 s -84 8 TIGLEY) 2 2 (4-4)3 7 (97–84 +1)253 +16) L{x} - s-4+8 (2-4) -) (3-4) L (3-4) + 3 2 (4-4)3 7 L{1} = 2 + (4-45 ñ 1 2. 1"/a-? + la i 1 elt 2. e4t - 4 Y(+) e e [Ano 12 is the required solution