You get t times t which is t squared plus t times b which is bt plus a times t which is at plus a times b which is ab.

So 1 times t. And v1 is minus 2 times t.

So it becomes t plus 3 times, times, t plus 5 times, I'll write that plus a little bit neater, times t plus 5.

So, this is t plus 3 times t plus 5. or t plus 5 times t plus 3 either way

You get 1 2t times the sine of t plus 1 4 sine of t times this thing in here, so times 2 sine of t cosine of t.

So if I factor a t out, that becomes t times t squared divided by t is t.

We're factoring out the t plus 3. t plus 3 times t, right?

So we get 4.9 times delta t squared 29.54 times delta t 10 is equal to zero

And all of that times t.

If you take t times t plus 3 and factor out the t plus 3, you're just left with that t.

This sine times this sine is sine squared of t cosine of t.

Delta t minus c and times dt.

Now with this said, this determinate becomers pretty simple. This becomes T times z two minus z1. this becomes T times...

It says 3 times, a plus b, t.

So these first two terms are the same thing as t times t plus 3.

This is f of t. e to the minus st times f of t dt.

That's the acceleration times time times the time that he's in the air, times this t.

I is equal to P times A times T population, affluence and technology.

But it was equal to the way we defined it x prime of t times i plus y prime of t times j.

So say some constant, c1, times my first function, f of t, plus some constant, c2, times my second function, g of t.

We get sine of t times the integral from 0 to t of this thing here.

And then over here I have minus 1 2 sine squared t times cosine of t.

So it's t times t plus 3 plus and then over here if you factor out a 5, you get a 5. times t plus 3, 5t divided by 5 is t, 15 divided by 5 is 3.

T is a transformation and it's B times x.

3 s times the Laplace transform of t squared.

So it'd be 3 times, a plus b, t.

When t is 0, cosine of 2 times 0.

All of that times 4.2 to the t, right?

So this is going to be minus 1 3 times the unit step function, where c is 2 pi of t times instead of sine of t sine of t minus 2pi.

The unit step function is zero until it gets to 2 times t, or of t, so, then it becomes 1 after t is equal to 2, times our function shifted by 2.

(t 1) times (z 2 minus z 1) is going to look like this. This is (t minus 1) times (z 2 minus z 1).

Or we could just rewrite this as 2 times the unit step function starting at 2, where that's when it becomes non zero of t times f of t minus 2. f of t minus 2 is this with t being replaced by t minus 2.

So it's this constant times my Dirac delta function, times delta of t minus c.

So cosine of t times sine of tau times the cosine of tau d tau.

So times how fast x changes with respect to t.

So, plus sin(t 2) times the j unit vector.

So we have 1 2 times the sine of t.

So times the Laplace transform of t to the 1.

This right here would be T times Z 2 minus Z 1. Now what would T minus 1 times Z 2 minus Z 1 look like?

The Laplace transform of t squared is equal to 2 s times the Laplace transform of t, of just t to the 1, right?

So it's 3 s times the Laplace transform of t to the n minus 1, so t squared.

So this is equal to the integral from 0 to t, of sine of t, times the cosine squared of tau d tau minus the integral from 0 to t of cosine of t times sine of tau cosine of tau d tau.

So let's just think about what happen if we multiply 2 binomials t plus a times t plus b

So I'm going to write the sine of t times the cosine of tau minus the sine of tau times the cosine of t, and then all of that's times the cosine of tau.

And y is equal to 2 times the sine of t.