(10 points) Find the formula for the Laplace transform of...

(10 points) Find the formula for the Laplace transform of the derivative of a function $y'$. Use the initial condition $y(0) = y_0$, use $y$ and $y'$ respectively for $y(t)$ and $y'(t)$ and $Y$ for the Laplace transform of $y$. $L\{y'(t)\}(s) = \int_0^\infty e^{-st}y'(t)dt$ $u = e^{-st}$ $dv = $ $du = [-se^{-st}]dt$ $v = $ $= $ =

Transcript

00:01 We have to find the laplus transformation for the post derivative that means l of y -dest t of s laplas transformation we have to do that means 0 to infinity e to the power minus of s t w -t that we have to find there are some information is given we can write y t as y you can write y -dest t as y -dast only and also it is given that y of zero is equal to y -0 and the laplas transformation of y is given as y so laplace transformation of y means e to the power zero to sorry integration zero to infinity e to the power minus of s t y of t d t is equal to laplous transformation that we will denote by the capital y so first you have to do the integration of this.

01:08 So for that we can use bipartisan method.

01:12 So for bipartisan method, we'll consider e to the power minus of st as you.

01:19 So then du will be minus of s, e to the power minus st.

01:25 Next, we'll consider dv is equal to y dash.

01:32 So then v will be y of t.

01:34 Y of t means only y we can write y of t that is equal to y so then let's do the integration so we have to write first of all into the power minus st this is the first function so you can write it as it is then when you do the integration of y -dust -t we will get y and we'll put the limit zero to infinity then minus of integration of a zero to infinity now already it contains minus we have to do the derivative of the first function and it already contains minus so minus minus become plus so it will be s e to the power minus of st then the integration of this that is y d t so then e to the power minus of s when we put the infinity so it will be e to the power minus infinity so one by e to the power infinity means very very well i mean small amount so we'll consider this as 0.

02:37 So 0 into y will be 0...

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