Question : Find a constant solution of y' = t(y - 1). Enter just an integer. : 2163497

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

1) Find a constant solution of y' = t(y - 1).

Enter just an integer.

2) Find a constant solution of y' = 10 y - 7.

Enter just a reduced fraction of form (a/b).

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

3) Consider the differential equation y' = t³(y + 3). Which of the following statements is/are true?

(I) f(t) = -3 is a constant solution to this differential equation.

(II) f(t) = 0 is a constant solution to this differential equation.

(III) If f(t) is a solution to the differential equation with initial conditions y(1) = 0, then f '(1) = 3.

A) II only

B) III only

C) I only

D) I, II, and III

E) I and III

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

4) Find f'(1) if f(t) is a solution to the initial value problem: y' = ty² + 5, y(1) = 1.

Enter just an integer.

5) Find f'(0) if f(t) is a solution to the initial value problem: y' = e^2t + y, y(0) = -1.

Enter just an integer.

6) Find f'(1) if f(t) is a solution to the initial value problem: y' = e^2t - y, y(1) = 0.

Enter just a real number (no approximations).

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

7) Consider the differential equation y' = y - y². Which of the following statements is/are true?

A) If f(t) is a solution to the differential equation satisfying the initial condition y(0) = 0, then f '(0) = 0.

B) This differential equation has infinitely many solutions.

C) The function f(t) = (1/(1 + e^-t)) is a solution to this differential equation with initial condition y(0) = (1/2).

D) The constant function f(t) = 1 is a solution to this differential equation.

E) All of these statements are true.

Solve the problem.

8) Write a differential equation that expresses the following description of a rate: When ice cream is removed from the freezer, it warms up at a rate proportional to the difference between the temperature of the ice cream and the room temperature of 76°. (Use y for the temperature of the ice cream, t for the time, and k for an unknown constant.)

A) y' = k(76 - t)

B) y' = 76 - ky

C) y' = 76t - ky

D) y' = k(76 - y)

9) The growth rate of a certain stock is modeled by (dV/dt) = k(44 - V), V = $21 when t = 0, where V = the value of the stock, per share, after time t (in months), and k = a constant. Find the solution to the differential equation in terms of t and k.

A) V = 44 - 44e^-kt

B) V = 44 - 23e^-kt

C) V = 21 - 23e^-kt

D) V = 44 - 23e^kt

10) Solve the differential equation model of radioactive decay:

(dQ/dt) = -0.3Q.

A) Q(t) = -Q₀ln0.3t + c

B) Q(t) = Q₀e^-t

C) Q(t) = (-1/0.3t) + Q₀

D) Q(t) = Q₀e^-0.3t

11) Sales (in thousands) of a certain product are declining at a rate proportional to the amount of sales, with a decay constant of 11% per year. Write a differential equation to express the rate of sales decline.

A) dy/dt = -0.11y

B) dy/dt = -0.89y

C) dy/dt = e^-0.11t

D) dy/dt = -0.11t

12) Which of the following functions solves the differential equation: y' = -4y?

A) y = - e^-4t

B) y = e^-4t

C) y = ln4t

D) none of these

13) Which of the following functions solves the differential equation: y' = e^-2x + 3?

A) y = e^-2x + 3

B) y = (1/2)e^-2x + 3

C) y = - (1/2)e^-2x + 3x

D) none of these

14) Which of the following functions solves the differential equation: y' = -6xy?

A) y = e^{-x^2}

B) y = 7e^{-x^2}

C) y = e^{-3x^2}

D) y = 7e^{-3x^2}

15) Which of the following functions solves the differential equation: y' = y²?

A) y = ln|1 + x|

B) y = (1/3)x³

C) y = - (1/x²)

D) y = - (1/x + 1)

Use the figure to answer the question.

16) The figure shows a slope field of the differential equation y' = 5y(5 - y). Use the figure to determine the constant solutions (if any) of the differential equation.

A) y = -5, y = 5

B) y = 0, y = 5

C) None

D) y = 0

17) The figure shows a slope field for the differential equation y' = t - y. Draw an approximation of a portion of the solution curve for y' = t - y that goes through the point (0, 2). Based on the slope field, can this solution pass through the point (1.1, 0.4)?