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Clicker Question 1

Clicker Question 1. Consider the DE y ' = 4 x . 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 = 5 C. y = 6 D. y = 8 E. y = 10. Separation of Variables (3/21/14).

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Clicker Question 1

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  1. Clicker Question 1 • Consider the DE y ' = 4x . 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 = 5 • C. y = 6 • D. y = 8 • E. y = 10

  2. Separation of Variables (3/21/14) • 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.

  3. 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 each, 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.

  4. Examples • Try the DE dy / dt = .08y . • Separate the variables: dy / y = .08 dt • Integrate both sides (the left with respect to y, the right with respect to t) and solve for y. • What solution satisfies the initial condition that y = 100 when t = 0? • Now try dy /dt = 4y2 • What solution satisfies the initial condition that y = 100 when t = 0?

  5. Clicker Question 2 • What is the general solution of dy/dx = xy ? • A. y = ½ x2 + C • B. y = e(1/2)x^2 + C • C. y = Ae(1/2)x^2 (A is any constant) • D. y = Ae(1/2)x^2 (A is a positive constant) • E. y = ln((1/2) x2) + C

  6. Clicker Question 3 • What is the general solution of dy / dx = y ? • A. y = x/ 2 + C • B. y = (x + C)2 / 4 • C. y = (x + C)2 / 2 • D. y = x2 / 4 + C • E. y = x2 / 2 + C

  7. 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?

  8. 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.

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

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