Solve using the Laplace transform: y" + 4y = g(t) where y(0) =
y'(0).
Hint: Use the convolution theorem to write your answer. You may
leave your answer expressed in terms of an integral.
Take the Laplace transform of the following initial value and
solve for Y(s)=L{y(t)}: y′′+4y={sin(πt) ,0, 0≤t<11≤t
y(0)=0,y′(0)=0
Y(s)= ? Hint: write the right hand side in
terms of the Heaviside function. Now find the inverse transform to
find y(t). Use step(t-c) for the Heaviside function u(t−c) .
y(t)= ?
Again considering y'' + 4y' + 3y = 0:
(a) Solve the IVP y'' + 4y' + 3y = 0; y(0) = 1, y'(0) = α where
α > 0.
(b) Determine the coordinates (tm,ym) of the maximum point of
the solution as a function of α.
(c) Determine the behavior of tm and ym as α →∞.