- #1
- 249
- 2
Homework Statement
Obtain the solution of the differential equation
x'' + w2nx = t
My use of L refers to the Laplace
Homework Equations
The Attempt at a Solution
L{x'' + w2nx = 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 w2nx is:
L{w2nx} = w2nx∫inf0e-st dt
because I believe that w2nx is a constant, hence I can move it out of the integral
evaluating the integral then gives me
L{w2nx} = w2nx/s
Then for L{t} I can see from the Laplce tables that this is just 1/s2
Consequently, my Laplace transformation is
L{x'' + w2nx = t} = sx - sx(0) - x'(0) + wn2x/s + 1/s2
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?