Back42–43. Implicit solutions for separable equations For the following separable equations, carry out the indicated analysis.
a. Find the general solution of the equation.
e⁻ʸᐟ²y'(x) = 4x sin x² − x; y(0) = 0, y(0) = ln(1/4), y(√(π/2)) = 0
{Use of Tech} Fish harvesting A fish hatchery has 500 fish at t=0, when harvesting begins at a rate of b>0fish/year The fish population is modeled by the initial value problem y′(t)=0.01y−b,y(0)=500 where t is measured in years.
c. Graph the solution in the case that b=60fish/year. Describe the solution.
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.
d. Identify the four regions in the first quadrant of the xy-plane in which x' and y' are positive or negative.
x′(t) = 2x − 4xy, y′(t) = −y + 2xy
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.
9. 2√(xy) (dy/dx) = 1, x, y > 0
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 and placed into a refrigerator whose temperature is constantly . After hour in the refrigerator, the pie is . What is the temperature of the pie 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.