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Differential Equations (10/14/13)

Differential Equations (10/14/13). A differential equation is an equation which contains derivatives within it.

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Differential Equations (10/14/13)

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  1. Differential Equations (10/14/13) • A differential equation is an equation which contains derivatives within it. • More specifically, it is an equation which may contain an independent variable x (or t) and/or a dependent variable y (or some other variable name), but definitely contains a derivativey ' = dy/dx (or dy/dt). • It may also contain second derivatives y '' , etc.

  2. Examples of DE’s • Every anti-derivative (i.e., indefinite integral) you have solved (or tried to solve) this semester is a differential equation! • What is y if y ' = x2 – 3x+ 5 ? • What is y if y ' = x / (x2 + 4) • What is y if dy/dt = e0.67t • Note that you also get a “constant of integration” in the solution.

  3. New types of examples • The following is a DE of a different type since it contains the dependent variable:y ' = .08y • Say in words what this says! Sound familiar? • Note that we don’t see the independent variable at all – let’s call it t . • What is a solution to this equation? And how can we find it?

  4. Clicker Question 1 • What is the most general solution of the differential equation 2 dy/dt = 5 / t4 ? • A. -5 / (3 t 3) + C • B. 5 / (3 t 3) + C • C. -5 / (6 t 3) + C • D. -1 / t 5 + C • E. -2 / t 5 + C

  5. The solutions to a DE • A solution of a given differential equation is a functiony which makes the equation work. • Show that y = Ae0.08t is a solution to the DE on the previous slide, where A is a constant. • Interpret this result! • Note that we are using the old tried and true method for solving equations here called “guess and check”.

  6. Examples of guess and check for DE’s • Show that y = 100 – A e –t satisfies the DE y ' = 100 - y • Show that y = sin(2t) satisfies the DE d2y / dt 2 = -4y • Show that y = x ln(x) – x satisfies the DEy ' = ln(x) • Of course one hopes for better methods to solve equations, but DE’s can be very hard.

  7. Clicker Question 2 • Which function below satisfies the DEd2y/dx2 = 9y ? • A. y = 3ex • B. y = 3sin(x) • C. y = sin(3x) • D. y = x2 + 9x • E. y = e3x

  8. Assignment for Wednesday • Read Section 9.1. • On page 584, do # 1 – 7 odd.

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