13. Intro to Differential Equations

[Use of Tech] Analysis of a separable equation Consider the differential equation yy'(t) = ½eᵗ + t and carry out the following analysis.

c. Graph the solutions in part (b) and describe their behavior as t increases. 

27–30. Predator-prey models Consider the following pairs of differential equations that model a predator-prey system with populations x and y. In each case, carry out the following steps.

e. Sketch a representative solution curve in the xy-plane and indicate the direction in which the solution evolves.

x′(t) = −3x + 6xy, y′(t) = y − 4xy

In Exercises 23–28, solve the initial value problem.

y dx + (3x - xy + 2)dy = 0, y(2) = -1, y < 0

Solve the initial value problems in Exercises 67–70 for x as a function of t.

(3t⁴ + 4t² + 1) (dx/dt) = 2√3, x(1) = -π√3/4

Solve the differential equation in Exercises 9–22.

11. (dy/dx) = e^(x-y)

Solve the differential equation in Exercises 9–22.

13. (dy/dx) = √y cos²√y

Solve the differential equation in Exercises 9–22.

17. (dy/dx) = 2x(y - 1), y > 1

Solve the differential equation in Exercises 9–22.

19. y²(dy/dx) = 3x²y³ - 6x²

In Exercises 125–128 solve the differential equation.

127. yy' = sec(y²)sec²(x)

In Exercises 1–22, solve the differential equation.

sec x dy + x cos² y dx = 0

In Exercises 1–22, solve the differential equation.

y' = xeˣ⁻ʸ csc y

In Exercises 1–22, solve the differential equation.

y' = (y²-1)x⁻¹

In Exercises 1–22, solve the differential equation.

y' = xy ln x ln y

In Exercises 43 and 44, let S represent the pounds of salt in a tank at time t minutes. Set up a differential equation representing the given information and the rate at which S changes. Then solve for S and answer the particular questions.

Pure water flows into a tank at the rate of 4 gal/min, and the well-stirred mixture flows out of the tank at the rate of 5 gal/min. The tank initially holds 200 gal of solution containing 50 pounds of salt.

b. How many pounds of salt are in the tank after 1 minute? after 30 minutes?

In Exercises 43 and 44, let S represent the pounds of salt in a tank at time t minutes. Set up a differential equation representing the given information and the rate at which S changes. Then solve for S and answer the particular questions.

Pure water flows into a tank at the rate of 4 gal/min, and the well-stirred mixture flows out of the tank at the rate of 5 gal/min. The tank initially holds 200 gal of solution containing 50 pounds of salt.

c. When will the tank have exactly 5 pounds of salt and how many gallons of solution will be in the tank?