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)= ?
Consider the following initial value problem.
y''−4y = 0,
y(0) = 0, y'(0) = 5
(a) Solve the IVP using the characteristic equation method from
chapter 4.
(b) Solve the IVP using the Laplace transform method from chapter
7.
(Hint: If you don’t have the same final answer for each part, you’ve
done something wrong.)
(3 pts) Solve the initial value problem
25y′′−20y′+4y=0, y(5)=0, y′(5)=−e2.
(3 pts) Solve the initial value problem
y′′ − 2√2y′ + 2y = 0, y(√2) = e2, y′(√2) = 2√2e2.
Consider the second order linear equation t2y′′+2ty′−2y=0,
t>0.
(a) (1 pt) Show that y1(t) = t−2 is a solution.
(b) (3 pt) Use the variation of parameters method to obtain a
second solution and a general solution.