In Exercise 1, we return to Exercises 2–5. (For convenience, the equations are reproduced...

In Exercise 1, we return to Exercises 2–5. (For convenience, the equations are reproduced below.) For each second-order equation,

(a) convert the equation to a first-order, linear system;

(b) compute the eigenvalues and eigenvectors of the system;

(c) for each eigenvalue, pick an associated eigenvector V, and determine the solution Y(t) to the system; and

(d) compare the results of your calculations in part (c) with the results that you obtained when you used the guess-and-test method

Exercise 1

Exercise 2

In Exercise, a harmonic oscillator equation for y (t) is given.

(a) Using HPGSystemSolver, sketch the associated direction field.

(b) Using the guess-and-test method described in this section, find two nonzero solutions that are not multiples of one another.

(c) For each solution, plot both its solution curve in the yv-plane and its y(t)- and v(t )-graphs.

Exercise 3

In Exercise, a harmonic oscillator equation for y (t) is given.

(a) Using HPGSystemSolver, sketch the associated direction field.

(b) Using the guess-and-test method described in this section, find two nonzero solutions that are not multiples of one another.

(c) For each solution, plot both its solution curve in the yv-plane and its y(t)- and v(t )-graphs

Exercise 4

In Exercise, a harmonic oscillator equation for y (t) is given.

(a) Using HPGSystemSolver, sketch the associated direction field.

(b) Using the guess-and-test method described in this section, find two nonzero solutions that are not multiples of one another.

(c) For each solution, plot both its solution curve in the y v-plane and its y(t)- and v(t )-graphs.

Exercise 5

In Exercise, a harmonic oscillator equation for y (t) is given.

(a) Using HPGSystemSolver, sketch the associated direction field.

(b) Using the guess-and-test method described in this section, find two nonzero solutions that are not multiples of one another.

(c) For each solution, plot both its solution curve in the yv-plane and its y(t)- and v(t )-graphs.