What are the properties of pollelogram?

Find two numbers that round to 264.5 when rounded to the nearest tenth

Vladimir79 [104]

Answer:

264.461 and also 264.549

6 0

3 years ago

Consider the initial value problemy' + 3 y = 36 t, y(0) = 6.Take the Laplace transform of both sides of the given differential e

wolverine [178]

Answer:

The solution of the differential equation is $y(t) = -4 + 12t + 10e^{-3t}$ .

Step-by-step explanation:

The first step is to take Laplace transform in both sides of the differential equation. As usual, we denote the Laplace transform of $y$ as $Y=L[y]$ . Then,

$L[y'+3y](s) = L[36t](s)$

$L[y'](s)+3L[y](s) = 36L[t](s)$

$sY(s)-y(0)+3Y(s) = \frac{36}{s^2}$

In the last step we use that $L[y'](s) = sL[y](s)-y(0)$ and $L[t](s) = \frac{1}{s^2}$ .

Notice that our differential equations becomes an algebraic equation for $Y(s)$ , which is more simple to solve.

In the expression we have obtained, we can write $Y(s)$ in terms of $s$ :

$(s+3)Y(s)-6=\frac{36}{s^2}$ which is equivalent to

$Y(s) = \frac{36}{s^2(s+3)} +\frac{6}{s+3}$ .

Now, we make a partial fraction decomposition for the term $\frac{36}{s^2(s+3)}$ . Thus,

$\frac{36}{s^2(s+3)} = -\frac{4}{s} + \frac{12}{s^2} + \frac{4}{s+3}$ .

Substituting the above value into the expression for $Y(s)$ we get

$Y(s) = -\frac{4}{s} + \frac{12}{s^2} + \frac{4}{s+3} + \frac{6}{s+3} = -\frac{4}{s} + \frac{12}{s^2} + \frac{10}{s+3}.Finally, we take the inverse Laplace transform (denoted by [tex]L^{-1}$ ) in both hands of the above expression. Recall that $y(t) = L^{-1}[Y](t)$ . So,