Circle the classifications which apply to each of the following differential equations (a) ty′ = y Separable Linear Nonlinear Exact (b) (2t + y) + (t − 6)dy dt = 0 Separable Linear Nonlinear Exact (c) y′ = cos(ty) Separable Linear Nonlinear Exact (d) y′ = t2 Separable Linear Nonlinear Exact (e) y′ = 2ty 1+y Separable Linear Nonlinear Exact
Consider the following slope field (and note that there are horizontal slopes (m = 0) along the x-axis, although they are difficult to see). (a) Draw and label 5 different solution curves on the slope field. (b) List the equilibrium solutions to the differential equation represented by this slope field: (c) Describe the behavior for each of the following initial conditions as we move forward in time (t): i. y(0) = 2 ii. y(0) = −1
Consider the initial value problem given by y′ = 2y−1 with the initial condition y(0) = 1. (a) Use Euler's method to find an approximate value of the solution at t = 2 using a step size of 0.5 (that is, Δt = 0.5 ). (b) Solve for the exact value of the solution at t = 2 by finding a particular solution to the original differential equation and then finding y(2). (c) In what way could we have altered the method in part (a) to get a solution value closer to the exact solution at t = 2?
Consider the following differential equation: (3ty + y2) + (t2 + ty)y′ = 0 (a) Show that the differential equation is not exact. (b) What is the integrating factor we can use to make this differential equation an exact differential equation? (c) Multiply through by the integrating factor, and then solve for the general solution. (Please leave your solution in explicit form, do not solve for y by itself [i.e. you don't need to write your solution as "y = . . . 1")
Draw a phase line for each of the following differential equations, and identify any and all equilibrium as stable, unstable, or semi-stable. (a) dy dt = (y−1)sin(y) (b) dy dt = y2+y−12
A certain coral reef is home to a population of hammerhead sharks. Following the implementation of a preservation program, scientists track the shark population over 10 years and obtain the following data: (a) Based on this data collection, approximate the carrying capacity of the population. (b) Assuming that this population follows the trend of a logistic growth model with a growth rate of 0.2, write a differential equation to represent the rate of change of number of sharks per year. dP dt = (c) Construct a phase line for your differential equation. (d) Does the carrying capacity you identified represent a stable unstable or semistable equilibrium solution?