Clicker Question 1 Which of the functions below might be one function shown in this direction field. A. y = (x – 1)2 B. y = 1 / x C. y = e-x D. y = sec(x) E. y = x2

Clicker Question 2 Consider the DE y ' = 2x . Using Euler’s method with x = 0.5 and starting at (0, 0), what is the estimate for y when x = 2 ? A. y = 4 B. y = 3.5 C. y = 3 D. y = 2.5 E. y = 2

Separation of Variables (10/18/13) Most differential equations are hard to solve exactly, i.e., it is hard to find an explicit description of a function y which satisfied the given DE. We can always try guess and check… We also learned about numerical / graphical techniques to get approximate solutions to DE’s.

A solution technique An exact technique which works on some DE’s is called separation of variables. The idea is that if you can completely separate the dependent and the independent variable from eachother, you can integrate each part separately. You must be able to put the equation in the form f (y ) dy = g (x) dx for some f and g.

Examples Again, every antiderivative problem y  = f (x) is separable. Previously we did the DE dy / dt = .08y . Now try dy /dt = 4y2 Separate the variables: dy / y2 = 4 dt Integrate both sides (the left with respect to y, the right with respect to t) What solution satisfies the “initial condition” that y = 100 when t = 0?

Clicker Question 3 What is the general solution to the DE dy/dx = y sin(x) ? A. y = -cos(x) + C (C is any constant) B. y = ln(cos(x) + C) C. y = 1 / (-cos(x) + C)2 D. y = A e-cos(x) (A is a positive constant) E. y = e-cos(x) + C

Another nice example: Consider the DE dy / dx = -x / y Think for a minute about what this says about the slopes on the graph of a solution function. Separate the variables, integrate both sides, and see what you get. What role does the constant of integration play in this case? What solution satisfies the initial condition that y = 6 when x = 0?

But most DE’s can’t be separated… This technique is no silver bullet. Being able to separate the variables is unusual. Example: dy / dx = x y + 10. Hmmm… It is not at all clear how to solve even a simple-looking one like this. We may need to turn to numerical/graphical techniques to get approximate solutions.

Assignment for Monday Read Section 9.3 In that section do Exercises 1-19 odd.