Laplace Transform of Step Function

ElijahRockers

1. The problem statement, all variables and given/known data

Solve

y'' + y = f(t), y(0)=0, y'(0)=1,

f(t)=
(0 for 0<t<pi)
(1 for pi<t<2pi)
(0 for t>2pi)

3. The attempt at a solution

y'' + y = u_pi(t)-u_2pi(t)

s²L{y} -sy(0) -y'(0) +L{y} = L{u_pi(t)} -Lu_2pi(t)}

L{y}(s²+1) -1 = (e^-pi*s/s) -(e^-2pi*s/s)

L{y} = (e^-pi*s/s(s²+1)) -(e^-2pi*s/s(s²+1)) +1/(s²+1)

This is where I get stuck... I'm assuming that I can factor out the e terms separately, then use decomposition of partial fractions to separate 1/s(s²+1), but when I do that I get meaningless values for A and B.

1/s(s²+1) = A/s + B/s²+1

1= A(s²+1) +Bs

1 = As² +Bs +A

From that I can infer that A = 1, but also that A=0, and B=0.

What am I doing wrong?

vela

The partial fraction expansion should be
$$\frac{1}{s(s^2+1)} = \frac{A}{s} + \frac{Bs+C}{s^2+1}$$ because the second term has a quadratic in the denominator.

ElijahRockers

Ah, thank you, I think I got it.

y = u_pi(t) -u_pi(t)cos(t-pi) -u_2pi(t) +u_2pi(t)cos(t-2pi) +sin(t)