- #1

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## Homework Statement

Obtain the solution of the differential equation

x'' + w

^{2}

_{n}x = t

My use of L refers to the Laplace

## Homework Equations

## The Attempt at a Solution

L{x'' + w

^{2}

_{n}x = t}

I decided to do the Laplace of each part individually starting with x''

L{x''} = sL{x'} - x'(0)

then

L{x'} = sL{x} - x(0)

Putting this together gives

L{x''} = s[sL{x} - x(0)] - x'(0)

Also, the L{x} = x/s

L{x''} = sx - sx(0) - x'(0)

Then the Laplace of w

^{2}

_{n}x is:

L{w

^{2}

_{n}x} = w

^{2}

_{n}x∫

^{inf}

_{0}e

^{-st}dt

because I believe that w

^{2}

_{n}x is a constant, hence I can move it out of the integral

evaluating the integral then gives me

L{w

^{2}

_{n}x} = w

^{2}

_{n}x/s

Then for L{t} I can see from the Laplce tables that this is just 1/s

^{2}

Consequently, my Laplace transformation is

L{x'' + w

^{2}

_{n}x = t} = sx - sx(0) - x'(0) + w

_{n}

^{2}x/s + 1/s

^{2}

After this step I'm pretty sure I need to do the inverse Laplace transform. But I wanted to check if what I had at the moment looked right, and if so, what is the best way to do the inverse of this laplace transformation?