1 / 7

6.3 Separation of Variables and the Logistic Equation

6.3 Separation of Variables and the Logistic Equation. Ex. 1 Separation of Variables. Find the general solution of. First, separate the variables. y’s on one side, x’s on the other. Second, integrate both sides. Take e to both sides. Solve for y. or.

keefer
Download Presentation

6.3 Separation of Variables and the Logistic Equation

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 6.3 Separation of Variables and the Logistic Equation Ex. 1 Separation of Variables Find the general solution of First, separate the variables. y’s on one side, x’s on the other. Second, integrate both sides. Take e to both sides. Solve for y. or

  2. Ex. 2 Finding a Particular Solution Given the initial condition y(0) = 1, find the particular solution of the equation To separate the variables, you must rid the first term of y and the second term of e-x^2. To do this, multiply both sides by ex^2/y. Now, integrate both sides.

  3. u = x2 du = 2x dx du/2x = dx Now find C at (0,1) or by multiplying by 2, you get

  4. Ex. 3 Finding a Particular Solution Curve Find the equation of the curve that passes through the point (1,3) and has a slope of y/x2 at any point (x,y). Because the slope is y/x2, you have Now, separate the variables. Take e to both sides to solve for y. at the point (1,3), C = ? or So, the equation is

  5. Ex. 4. Wildlife Population The rate of change of the number of coyotes N(t) in a population is directly proportional to 650 - N(t), where t is the time in years. When t = 0, the population is 300, and when t = 2, the population has increased to 500. Find the population when t = 3. Because the rate of change of the population is proportional to 650 - N(t), we can write the following differential equation. Separate variables Integrate

  6. Take e to both sides. Using N = 300 when t = 0, you can conclude that C = 350, which produces Then, using N = 500 when t = 2, it follows that So, the model for the coyote population is When t = 3, the approximate population is

More Related