13. Intro to Differential Equations

In Exercises 1–22, solve the differential equation.

y' = sin³ x cos² 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?

Solve the following initial value problem for u as a function of t:

du/dt + (k/m) u = 0 (k and m positive constants), u(0) = u₀

b. as a separable equation.

Solve the differential equation in Exercises 9–22.

10. (dy/dx) = x²√y, y > 0

Solve the differential equation in Exercises 9–22.

12. (dy/dx) = 3x²e^(-y)

In Exercises 1–22, solve the differential equation.

y' = xeʸ√(x-2)

In Exercises 1–22, solve the differential equation.

y' = eʸ/xy

Show that (0, 0) and (c/d, a/b) are equilibrium points. Explain the meaning of each of these points.

Carbon monoxide pollution An executive conference room of a corporation contains 4500 ft³ of air initially free of carbon monoxide. Starting at time t = 0, cigarette smoke containing 4% carbon monoxide is blown into the room at the rate of 0.3 ft³/min. A ceiling fan keeps the air in the room well circulated and the air leaves the room at the same rate of 0.3 ft³/min. Find the time when the concentration of carbon monoxide in the room reaches 0.01%.

Solve the homogeneous equations in Exercises 5–10. First put the equation in the form of a homogeneous equation.

(x²+y²)dx + xy dy = 0

Solve the homogeneous equations in Exercises 5–10. First put the equation in the form of a homogeneous equation.

(x.exp(y/x) + y)dx - x dy = 0

Solve the homogeneous equations in Exercises 5–10. First put the equation in the form of a homogeneous equation.

y' = y/x + cos ((y-x)/x)