We established that the inverse Laplace transform of that big thing that I had written before, 2 times s minus 1 times e to the minus 2 sorry, e to the minus 2s over s squared minus 2s plus 2 is equal to this thing where f of t is this.

Multiply it times our original differential equation.

Let's do the inverse Laplace transform of the whole thing.

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What we learn is that if it can be homogeneous, if this is a homogeneous differential equation, that we can make a variable substitution.

But we're done, we factored it.

let's ignore the 2.

We know the exponential function.

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So let's see if we can use this with this information to get a generalized formula.

So using the derivative property of Laplace transform, we figured out the Laplace transform of cosine of a t and the Laplace transform of really any polynomial, right?

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The answer is 200 mm for our focal length.

And this is actually going to be a big clue as to why

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But as we will see we can simplify this to turn it into a linear equation.

Thrun And the answer is 3200cm or 32 meters.

Interesting.

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