13. Intro to Differential Equations

Solve the differential equation in Exercises 9–22.

19. y²(dy/dx) = 3x²y³ - 6x²

23–26. Stirred tank reactions For each of the following stirred tank reactions, carry out the following analysis.

b. Solve the initial value problem.

A one-million-liter pond is contaminated by a chemical pollutant with a concentration of 20 g/L. The source of the pollutant is removed, and pure water is allowed to flow into the pond at a rate of 1200 L/hr. Assuming the pond is thoroughly mixed and drained at a rate of 1200 L/hr, how long does it take to reduce the concentration of the solution in the pond to 10% of the initial value?

A pie is removed from an oven and its temperature is $175\text{℃}$ and placed into a refrigerator whose temperature is constantly $3\text{℃}$. After $1$ hour in the refrigerator, the pie is $90\text{℃}$. What is the temperature of the pie $4$ hours after being placed in the refrigerator?

Solve the differential equation in Exercises 9–22.

10. (dy/dx) = x²√y, y > 0

5–10. First-order linear equations Find the general solution of the following equations.

v'(y) − v/2 = 14

Orthogonal trajectories Use the method in Exercise 44 to find the orthogonal trajectories for the family of circles x² + y² = a²

20–22. {Use of Tech} Solving the Gompertz equation Solve the Gompertz equation in Exercise 19 with the given values of r, K, and M₀. Then graph the solution to be sure that M(0) and lim(t→∞) M(t) are correct.

r = 0.05, K = 1200, M₀ = 90

In Exercises 1–22, solve the differential equation.

y' = eʸ/xy

In Exercises 1–22, solve the differential equation.

y' = xy ln x ln y

[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. 

11–18. Solving initial value problems Use the method of your choice to find the solution of the following initial value problems.

y′(x) = 4x csc y, y(0) = π/2

Solve the differential equation in Exercises 9–22.

13. (dy/dx) = √y cos²√y

Find the particular solution that satisfies the given initial condition

$(y^2+y)e^x{y}^{\prime }=y^3+e^x{y}^3;y\left(0\right)=1$

Case 2 of the general solution Solve the equation y′(t) = ky + b in the case that ky + b < 0 and verify that the general solution is y(t) = Ceᵏᵗ − b/k.

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