Callahan:  wk 7
11.3: 7, 8, 13, 15, 16, 17
11.4: 1, 2, 6, 7, 10

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11.3: 7, 8, 13, 15, 16, 17     `` ``
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11.4: 1, 2, 6, 7, 10

1)  Use the method of separation of variables to find a formula for the solution of the differential equation .  Your formula should contain an arbitrary constant to reflect the fact that many functions solve the differential equation.

First separate variables:

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Now we integrate to solve the differential equation for :

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Here we need to use a substitution to continue:

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So our general solution to the differential equation is:

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6)  A falling body with air resistance.  We have used the differential equation:

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to model the motion of a body falling under the influence of gravity and air resistance .  Here is the velocity of the body at time .

a) Solve the differential equation by seperating variables, and obtain:

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First separate variables:

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Now we integrate to solve the differential equation for :

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Here we need to use a substitution to continue:

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So our general solution to the differential equation is:

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b) Now impose the initial condition to determine the value of :

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c) Show that your solution to the initial value problem is equivilent to the answer found in exercise 21 (p197):

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So the solutions are equivilent.

d) Knowing that and , the distance that the body has fallen by time is given by the integral:

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Use your formula for from part (b) to find :

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Here we need to use a substitution to continue:

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7)  Supergrowth:

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10)  Find a formula for each of these indefinite integrals using partial fractions:

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So we know that all of the coefficients must equal zero so that the whole polynomial equals zero:

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Here we need to use some substitutions to continue:

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`` ``  Converted by Mathematica      May 24, 2004