Help with Laplace Transformations and 2nd order ODEs

[TFM]

Okay so:

$$Y = \frac{\frac{1}{s^2 + 1} + s - 1/2}{s^2 + 1}$$

$$Y = \frac{1}{(s^2 + 1)^2} + \frac{s}{s^2 + 1} - \frac{1}{2s^2 + 2}$$

So, the Laplace tranformation of:

$$\frac{1}{(s^2 + 1)^2} = = \frac{1}{2} (-sin(t)- tcos(t))$$

and the Laplace transformation of:

$$\frac{s}{s^2+1} = cos(t)$$

And finally, the Laplace Transformation of:

$$\frac{1}{2s^2 + 2}$$

would this be similar to the first one, but require first shift theorem:

$$\frac{1}{(s^2 + 1)^2} = = \frac{1}{2} (-sin(t)- tcos(t))$$

$$\frac{1}{(2s^2 + 2)^2} = = \left(\frac{1}{2} (-sin(t)- tcos(t))\right) e^2t$$

Is this okay?

TFM

 

[sara_87]

The first two bits are perfectly fine.
what is the inverse laplace transform of 1/(s^2 +1)
It is ofcourse sin(t)
so that means the inverse laplace of 1/2(s^2+1) is just simply (1/2)sin(t)

so now you have the inverse laplace of each bit, now you can state what the function y(t) is.

 

[TFM]

Okay, so now:

$$y(t) = cos(t) + \frac{1}{2}(-sin(t) - t cos(t)) + \frac{1}{2}sin(t)$$

Is this okay?

TFM

 

[sara_87]

that's right

 

[TFM]

Excellent. Thanks for all your assistance, sara_87