Written homework #5.
Problem #1: Find the general solution of the following system of equations and describe the behavior of the solution
as t ! 1. Plot a few trajectories of this system.
Problem #2: Find the general solution of the following system of equations and describe the behavior of the solution as t ! 1. Plot a few trajectories of this system.
Problem #3: Find the solution of the following initial value problem and describe the behavior of the solution as
Problem #4: Find the general solution of the following system of equations and describe the behavior of the solution as t ! 1. Plot a few trajectories of this system.
Problem #5: Find the Laplace transform of the function f(t) = 4e
Problem #6: Find the inverse Laplace transform of the function F(s) =
5e
Problem #7: Find the inverse Laplace transform of the function F(s) =
Problem #8: Use the Laplace transform to solve the following initial value problem.
Problem #9: Use the Laplace transform to solve the following initial value problem.
Problem #10: Use the Laplace transform to solve the following initial value problem.
Problem #11: Use the Laplace transform to solve the following initial value problem.
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
