Question

solution of IVP using laplace transform.
(D^2 + 2D +1)y = e^(-1) +sinx
y(0)=1 y'(0)=0

Answer 1

Is D suppose to be derivative?

Answer 2

no. it's just a variable :)

Answer 3

Uh, I'm pretty sure it's the derivative otherwise you have 3 variables which is weird

Answer 4

So all you do is take the laplace of the left hand side and the right hand side going term by term. So the laplace of y'' is Ys^2-y(0)s-y'(0) and the laplace of y'=Ys-y(0) and the laplace of y is just Y. Now for the right hand side, you e^-1 is a constant, and laplace transforms of constants is just 1/C where C is the constant. So the laplace transfrom of e^-1 is just e. You can get the transfrom of sinx from your laplace table. After doing that, simply plug in your values and rearrange to get thigns nice and clean. Then isolate for Y, use partial fractions to split up the giant fraction you get so you can then inverse laplace everything using your table to get back little y.