find the response of the function at * (t=4) using Laplace transform (y() + 2y" + y = sint) y(0)=1, y (0)=-2, y"(0)=3 , y"(0)=0

Considering you have four initial conditions (the last of which should probably read ), I'm assuming the ODE is

with , , , and .

Take the Laplace transform of both sides, denoting the transform of by :

Solve for :

Notice that

and simplify a bit to get

Decompose into partial fractions:

So we have

Split up the first term to get two easy inverse transforms:

Also split up the second term, but use the convolution theorem, which says

where and are the Laplace transforms of and , respectively, and the convolution is defined by

Take

so that

and their convolution is

Next, take

You can treat the third term similarly, but with an extra step. First compute

by taking

Then

Next, take

Thus we end up with the solution,

kjgfcvbjkl

step-by-step explanation:

sdfghjkl; vbnm,