Back27–30. 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.
9. 2√(xy) (dy/dx) = 1, x, y > 0
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.
15. √x (dy/dx) = e^(y+√x), x > 0
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
Solve the differential equation in Exercises 9–22.
21. (1/x)(dy/dx) = ye^(x²) + 2√y e^(x²)
In Exercises 125–128 solve the differential equation.
125. dy/dx = √y cos(√y)
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' = sin³ x cos² y