Laplace transform of y''+y=e^(2t)?

Question

Hey,

I am trying to find laplace transform of y''+y=e^(2t) and so far I got:

Y(s) = (s-1)/((s-2)(s^2+1))

what do i do next?

Answer 1

I'm not SURE what you wrote is correct (it's been a while since I have done Laplace Transforms) but if what you have written is correct, you will now most likely want to split the RHS into partial fractions so that you have:

Y(s) = A(s)/(s-2) + B(s)/(s^2 + 1) and then from these, you should be able to do the inverse Laplace transform to get back to a solution in y.

To do this, I would guess that A(s) = a and B(s) = bs + c

Then if you plug in and set the two equations equal, you get

a(s^2 + 1) + (bs+c)(s-2) = s - 1.

Multiplying out and setting the s^i terms equal to each other gives that:

a + b = 0
a - 2c = -1
c - 2b = 1

which gives me a = 1/5, b = -1/5 and c = 3/5 (though you may want to check this).

In general, partial fractions are good thing to try when it comes to Laplace Transform problems (since they tend to occur frequently).