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Differential Equations (4/17/06) 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 derivative y ' = dy/dx (or dy/dt). It may also contain second derivatives y '', etc.

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Examples of DE’s Every anti-derivative (i.e., indefinite integral) you have solved (or tried to solve) this year is a differential equation! What is y if y ' = x 2 – 3x + 5 ? What is y if y ' = x / (x 2 + 4) What is y if dy/dt = e 0.67t Note that you also get a “constant of integration” in the solution.

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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?

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The solutions to a DE A solution of a given differential equation is a function y which makes the equation work. Show that y = Ae 0.08t 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”.

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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 d 2 y / dt 2 = -4y Show that y = x ln(x) – x satisfies the DE y ' = ln(x) Of course one hopes for better methods to solve equations, but DE’s can be very hard.

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Assignment for Wednesday Go through the steps which the calculator does to compute 123.45 0.654. How do you think your calculator computes sin(23.56)? Read Section 9.1 On page 591, do # 1 – 9 odd.

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