Answer: In each of 1 through 8: G a. Draw a direction field for the given differential

In each of Problems 1 through 8: G a. Draw a direction field for the given differential equation. b. Based on an inspection of the direction field, describe how solutions behave for large t. c. Find the general solution of the given differential equation, and use it to determine how solutions behave as t . y_ + 3y = t + e2t

In each of Problems 1 through 8: G a. Draw a direction field for the given differential equation. b. Based on an inspection of the direction field, describe how solutions behave for large t. c. Find the general solution of the given differential equation, and use it to determine how solutions behave as t . y_ 2y = t2e2t

In each of Problems 1 through 8: G a. Draw a direction field for the given differential equation. b. Based on an inspection of the direction field, describe how solutions behave for large t. c. Find the general solution of the given differential equation, and use it to determine how solutions behave as t . y_ + y = tet + 1

In each of Problems 1 through 8: G a. Draw a direction field for the given differential equation. b. Based on an inspection of the direction field, describe how solutions behave for large t. c. Find the general solution of the given differential equation, and use it to determine how solutions behave as t . y_ +1t y = 3 cos(2t), t > 0

In each of Problems 1 through 8: G a. Draw a direction field for the given differential equation. b. Based on an inspection of the direction field, describe how solutions behave for large t. c. Find the general solution of the given differential equation, and use it to determine how solutions behave as t . y 2y = 3et

In each of Problems 1 through 8: G a. Draw a direction field for the given differential equation. b. Based on an inspection of the direction field, describe how solutions behave for large t. c. Find the general solution of the given differential equation, and use it to determine how solutions behave as t . ty_ y = t2et , t > 0

In each of Problems 1 through 8: G a. Draw a direction field for the given differential equation. b. Based on an inspection of the direction field, describe how solutions behave for large t. c. Find the general solution of the given differential equation, and use it to determine how solutions behave as t . y_ + y = 5 sin(2t)

In each of Problems 1 through 8: G a. Draw a direction field for the given differential equation. b. Based on an inspection of the direction field, describe how solutions behave for large t. c. Find the general solution of the given differential equation, and use it to determine how solutions behave as t . 2y_ + y = 3t2

In each of Problems 9 through 12, find the solution of the given initial value problem. y_ y = 2te2t , y(0) = 1

In each of Problems 9 through 12, find the solution of the given initial value problem. y_ + 2y = te2t , y(1) = 0

In each of Problems 9 through 12, find the solution of the given initial value problem. y_ +2t y = cos t t2 , y( ) = 0, t > 0

In each of Problems 9 through 12, find the solution of the given initial value problem. ty_ + (t + 1) y = t, y(ln 2) = 1, t > 0

In each of Problems 13 and 14: G a. Draw a direction field for the given differential equation. How do solutions appear to behave as t becomes large? Does the behavior depend on the choice of the initial value a? Let a0 be the value of a for which the transition from one type of behavior to another occurs. Estimate the value of a0. b. Solve the initial value problem and find the critical value a0 exactly. c. Describe the behavior of the solution corresponding to the initial value a0. y_ 1 2 y = 2 cos t, y(0) = a

In each of Problems 13 and 14: G a. Draw a direction field for the given differential equation. How do solutions appear to behave as t becomes large? Does the behavior depend on the choice of the initial value a? Let a0 be the value of a for which the transition from one type of behavior to another occurs. Estimate the value of a0. b. Solve the initial value problem and find the critical value a0 exactly. c. Describe the behavior of the solution corresponding to the initial value a0. 3y_ 2y = e t/2, y(0) = a

In each of Problems 15 and 16: G a. Draw a direction field for the given differential equation. How do solutions appear to behave as t 0? Does the behavior depend on the choice of the initial value a? Let a0 be the critical value of a, that is, the initial value such that the solutions for a < a0 and the solutions for a > a0 have different behaviors as t . Estimate the value of a0. b. Solve the initial value problem and find the critical value a0 exactly. c. Describe the behavior of the solution corresponding to the initial value a0. ty_ + (t + 1) y = 2tet , y(1) = a, t > 0

In each of Problems 15 and 16: G a. Draw a direction field for the given differential equation. How do solutions appear to behave as t 0? Does the behavior depend on the choice of the initial value a? Let a0 be the critical value of a, that is, the initial value such that the solutions for a < a0 and the solutions for a > a0 have different behaviors as t . Estimate the value of a0. b. Solve the initial value problem and find the critical value a0 exactly. c. Describe the behavior of the solution corresponding to the initial value a0. (sin t) y_ + (cos t) y = et , y(1) = a, 0 < t < G

Consider the initial value problem y_ + 1 2 y = 2 cos t, y(0) = 1. Find the coordinates of the first local maximum point of the solution for t > 0. N

Consider the initial value problem y_ + 2 3 y = 1 1 2 t, y(0) = y0. Find the value of y0 for which the solution touches, but does not cross, the t-axis.

Consider the initial value problem y_ + 1 4 y = 3 + 2 cos(2t), y(0) = 0. a. Find the solution of this initial value problem and describe its behavior for large t. N b. Determine the value of t for which the solution first intersects the line y = 12.

Find the value of y0 for which the solution of the initial value problem y_ y = 1 + 3 sin t, y(0) = y0 remains finite as t .

Consider the initial value problem y_ 3 2 y = 3t + 2et , y(0) = y0. Find the value of y0 that separates solutions that grow positively as t from those that grow negatively. How does the solution that corresponds to this critical value of y0 behave as t ?

Show that all solutions of 2y_ + ty = 2 [equation (41) of the text] approach a limit as t , and find the limiting value. Hint: Consider the general solution, equation (47). Show that the first term in the solution (47) is indeterminate with form 0 . Then, use lHpitals rule to compute the limit as t .

Show that if a and are positive constants, and b is any real number, then every solution of the equation y_ + ay = bet has the property that y 0 as t . Hint: Consider the cases a = and a _= separately.

All solutions have the limit 3 as t .

All solutions are asymptotic to the line y = 3 t as t .

All solutions are asymptotic to the line y = 2t 5 as t .

All solutions approach the curve y = 4 t2 as t .

Variation of Parameters. Consider the following method of solving the general linear equation of first order: y_ + p(t) y = g(t). (48) a. If g(t) = 0 for all t, show that the solution is y = A exp__ p(t) dt_, (49) where A is a constant. b. If g(t) is not everywhere zero, assume that the solution of equation (48) is of the form y = A(t) exp__ p(t) dt_, (50) where A is now a function of t. By substituting for y in the given differential equation, show that A(t) must satisfy the condition A_(t) = g(t) exp__ p(t) dt_ . (51) c. Find A(t) from equation (51). Then substitute for A(t) in equation (50) and determine y. Verify that the solution obtained in this manner agrees with that of equation (33) in the text. This technique is known as the method of variation of parameters; it is discussed in detail in Section 3.6 in connection with secondorder linear equations.

In each of Problems 29 and 30, use the method of Problem 28 to solve the given differential equation. y_ 2y = t2e2t

In each of Problems 29 and 30, use the method of Problem 28 to solve the given differential equation. y_ +1t y = cos(2t), t > 0