See the attached file for questioins

Written homework #5.

Problem #1: Find the general solution of the following system of equations and describe the behavior of the solution as t →∞. Plot a few trajectories of this system.

x′ =

[ −4 3 −2 1

] x

Problem #2: Find the general solution of the following system of equations and describe the behavior of the solution as t →∞. Plot a few trajectories of this system.

x′ =

[ −1 1 −4 3

] x

Problem #3: Find the solution of the following initial value problem and describe the behavior of the solution as t →∞.

x′ =

[ 4 2 3 −1

] x, x(0) =

[ 2 3

] Problem #4: Find the general solution of the following system of equations and describe the behavior of the solution as t →∞. Plot a few trajectories of this system.

x′ =

[ −1 1 −4 −1

] x

Problem #5: Find the Laplace transform of the function f(t) = 4e−5t + 2 cos 3t + 5u4(t)−1.

Problem #6: Find the inverse Laplace transform of the function F(s) = 5e−2s

s2 −4 .

Problem #7: Find the inverse Laplace transform of the function F(s) = 3s

s2 −s−6 .

Problem #8: Use the Laplace transform to solve the following initial value problem.

y′′ + 3y′ + 2y = 0, y(0) = 1, y′(0) = 0

Problem #9: Use the Laplace transform to solve the following initial value problem.

y′′ + 2y′ + 5y = 0, y(0) = 2, y′(0) = −1

Problem #10: Use the Laplace transform to solve the following initial value problem.

y′′ + 5y′ + 6y = u3(t), y(0) = 1, y ′(0) = 0

Problem #11: Use the Laplace transform to solve the following initial value problem.

y′′ + 2y′ + 2y = g(t), y(0) = 0, y′(0) = 1, and where

g(t) =

{ 1, π ≤ t < 2π 0, 0 ≤ t < π and t ≥ 2π

1