Figure 11.42 shows the slope field for dy/dx = y2. (a) Sketch the solutions that pass

Determine which of the following differential equations are separable. Do not solve the equations. (a) y = y (b) y = x + y (c) y = xy (d) y = sin(x + y) (e) y xy = 0 (f) y = y/x (g) y = ln (xy) (h) y = (sin x)(cos y) (i) y = (sin x)(cos xy) (j) y = x/y (k) y = 2x (l) y = (x+y)/(x+ 2y)

In Exercises 228,use separation of variables to find the solutions to the differential equations subject to the given initial conditions.

In Exercises 228,use separation of variables to find the solutions to the differential equations subject to the given initial conditions.

In Exercises 228,use separation of variables to find the solutions to the differential equations subject to the given initial conditions.

In Exercises 228,use separation of variables to find the solutions to the differential equations subject to the given initial conditions.

In Exercises 228,use separation of variables to find the solutions to the differential equations subject to the given initial conditions.

In Exercises 228,use separation of variables to find the solutions to the differential equations subject to the given initial conditions.

In Exercises 228,use separation of variables to find the solutions to the differential equations subject to the given initial conditions.

In Exercises 228,use separation of variables to find the solutions to the differential equations subject to the given initial conditions.

In Exercises 228,use separation of variables to find the solutions to the differential equations subject to the given initial conditions.

In Exercises 228,use separation of variables to find the solutions to the differential equations subject to the given initial conditions.

In Exercises 228,use separation of variables to find the solutions to the differential equations subject to the given initial conditions.

In Exercises 228,use separation of variables to find the solutions to the differential equations subject to the given initial conditions.

In Exercises 228,use separation of variables to find the solutions to the differential equations subject to the given initial conditions.

In Exercises 228,use separation of variables to find the solutions to the differential equations subject to the given initial conditions.

In Exercises 228,use separation of variables to find the solutions to the differential equations subject to the given initial conditions.

In Exercises 228,use separation of variables to find the solutions to the differential equations subject to the given initial conditions.

In Exercises 228,use separation of variables to find the solutions to the differential equations subject to the given initial conditions.

In Exercises 228,use separation of variables to find the solutions to the differential equations subject to the given initial conditions.

In Exercises 228,use separation of variables to find the solutions to the differential equations subject to the given initial conditions.

In Exercises 228,use separation of variables to find the solutions to the differential equations subject to the given initial conditions.

In Exercises 228,use separation of variables to find the solutions to the differential equations subject to the given initial conditions.

In Exercises 228,use separation of variables to find the solutions to the differential equations subject to the given initial conditions.

In Exercises 228,use separation of variables to find the solutions to the differential equations subject to the given initial conditions.

In Exercises 228,use separation of variables to find the solutions to the differential equations subject to the given initial conditions.

In Exercises 228,use separation of variables to find the solutions to the differential equations subject to the given initial conditions.

In Exercises 228,use separation of variables to find the solutions to the differential equations subject to the given initial conditions.

In Exercises 228,use separation of variables to find the solutions to the differential equations subject to the given initial conditions.

(a) Solve the differential equation dy dx = 4x y2 . Write the solution y as an explicit function of x. (b) Find the particular solution for each initial condition below and graph the three solutions on the same coordinate plane. y(0) = 1, y(0) = 2, y(0) = 3.

(a) Solve the differential equation dP dt = 0.2P 10. Write the solution P as an explicit function of t. (b) Find the particular solution for each initial condition below and graph the three solutions on the same coordinate plane. P(0) = 40, P(0) = 50, P(0) = 60

(a) Find the general solution to the differential equation modeling how a person learns: dy dt = 100 y. (b) Plot the slope field of this differential equation and sketch solutions with y(0) = 25 and y(0) = 110. (c) For each of the initial conditions in part (b), find the particular solution and add to your sketch. (d) Which of these two particular solutions could represent how a person learns?

A circular oil spill grows at a rate given by the differential equation dr/dt = k/r, where r represents the radius of the spill in feet, and time is measured in hours. If the radius of the spill is 400 feet 16 hours after the spill begins, what is the value of k? Include units in your answer

Figure 11.42 shows the slope field for dy/dx = y2. (a) Sketch the solutions that pass through the points (i) (0, 1) (ii) (0, 1) (iii) (0, 0) (b) In words, describe the end behavior of the solution curves in part (a). (c) Find a formula for the general solution. (d) Show that all solution curves except for y = 0 have both a horizontal and a vertical asymptote.

Solve the differential equations in Problems 3443. Assume a, b, and k are nonzero constants.

Solve the differential equations in Problems 3443. Assume a, b, and k are nonzero constants.

Solve the differential equations in Problems 3443. Assume a, b, and k are nonzero constants.

Solve the differential equations in Problems 3443. Assume a, b, and k are nonzero constants.

Solve the differential equations in Problems 3443. Assume a, b, and k are nonzero constants.

Solve the differential equations in Problems 3443. Assume a, b, and k are nonzero constants.

Solve the differential equations in Problems 3443. Assume a, b, and k are nonzero constants.

Solve the differential equations in Problems 3443. Assume a, b, and k are nonzero constants.

Solve the differential equations in Problems 3443. Assume a, b, and k are nonzero constants.

Solve the differential equations in Problems 3443. Assume a, b, and k are nonzero constants.

Solve the differential equations in Problems 4447. Assume x, y, t > 0.

Solve the differential equations in Problems 4447. Assume x, y, t > 0.

Solve the differential equations in Problems 4447. Assume x, y, t > 0.

Solve the differential equations in Problems 4447. Assume x, y, t > 0.

Figure 11.43 shows the slope field for the equation dy dx =  y2 if |y| 1 1 if 1 y 1. (a) Sketch the solutions that pass through (0, 0). (b) What can you say about the end behavior of the solution curve in part (a)? (c) For each of the following regions, find a formula for the general solution (i) 1 y 1 (ii) y 1 (iii) y 1 (d) Show that each solution curve has two vertical asymptotes.

(a) Sketch the slope field for y = x/y. (b) Sketch several solution curves. (c) Solve the differential equation analytically

(a) Sketch the slope field for y = y/x. (b) Sketch several solution curves. (c) Solve the differential equation analytically.

Compare the slope field for y = x/y, Problem 49, with that for y = y/x, Problem 50. Show that the solution curves of Problem 49 intersect the solution curves of Problem 50 at right angles.

Separating variables in dy/dx = x + y gives y dy = x dx.

The solution to dP/dt = 0.2t is P = Be0.2t

Separating variables in dy/dx = ex+y gives ey dy = ex dx.

A differential equation that is not separable.

An expression for f(x) such that the differential equation dy/dx = f(x) + xy cos x is separable

A differential equation all of whose solutions form the family of functions f(x) = x2 + C.

A differential equation all of whose solutions form the family of hyperbolas x2 y2 = C.

For all constants k, the equation y + ky = 0 has exponential functions as solutions.

The differential equation dy/dx = x + y can be solved by separation of variables.

The differential equation dy/dx xy = x can be solved by separation of variables.

The only solution to the differential equation dy/dx = 3y2/3 passing through the point (0, 0) is y = x3.

Match the graphs in Figure 11.50 with the following descriptions. (a) The temperature of a glass of ice water left on the kitchen table. (b) The amount of money in an interest-bearing bank account into which $50 is deposited. (c) The speed of a constantly decelerating car. (d) The temperature of a piece of steel heated in a furnace and left outside to cool.