Consider the initial value problem y” + 9y = 27t, y(0) = 7,...

Consider the initial value problem y'' + 9y = 27t, y(0) = 7, y'(0) = 9. a. Take the Laplace transform of both sides of the given differential equation to create the corresponding algebraic equation. Denote the Laplace transform of y(t) by Y(s). Do not move any terms from one side of the equation to the other (until you get to part (b) below). = help (formulas) b. Solve your equation for Y(s). Y(s) = L {y(t)} = c. Take the inverse Laplace transform of both sides of the previous equation to solve for y(t). y(t) =

Transcript

00:01 All right, we're going to solve this differential equation using laplace transforms.

00:05 So i'm going to take the laplace transform of each side, which is equivalent to taking the laplace transform of each term.

00:14 Okay, and the nine can go out in the front here.

00:28 All right, so first, we need the laplace transformer of y double prime.

00:33 So here it is.

00:36 So you put s squared, y of s minus s.

00:43 Y of 0 minus y prime of 0 plus 9.

00:50 And now we need the laplace transform of y, which is just y of s.

01:04 And that's equal to 27 times the laplace transform of d, which is 1 over s squared.

01:18 All right.

01:19 So i'm going to put these two together.

01:23 So we have y of s times s squared.

01:27 Plus 9 minus s times y of 0 and y of 0 is 7 so 7 s minus y prime of 0 which is 9.

01:39 Oops, i already did that and that's equal to 27 over s squared.

01:50 So i have y of s, s squared plus 9 equals 27 over s squared plus 9 plus 27 plus 7 s plus 9.

02:07 And then we're going to divide the s squared plus 9 over there.

02:11 But before we do, i'm going to have to do partial fractions anyway.

02:19 So i think i'm just going to go ahead and get a common denominator here and add these together.

02:26 So i get y of s equals 27 plus 7s cubed plus 9 s squared, all over s squared, which is this s squared.

02:40 And then s squared plus 9.

02:46 Okay, so that's why this.

02:49 And now what you have to do is partial fractions so that we can take it apart and then use the table backwards and do inverse transforms.

02:59 All right, so we have a over s plus b over s squared plus c s plus d over s squared plus 9.

03:14 And this is equal to 7s cubed plus 9 s squared plus 27.

03:25 So as times s squared plus 9 plus b times s squared plus 9 plus c s plus d times s squared 7 s cubed plus 9 s squared plus 27 as cubed plus 9a s squared plus 9 as cubed plus 9a s squared plus 9b plus 9b plus plus c s cubed plus d s squared.

04:09 All right, so we have a s cubed, c s cube equals 7s cubed.

04:15 So a plus c equals seven...

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