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

11.3: 7, 8, 13, 15, 16, 17

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:

    Now we integrate to solve the differential equation for :

    Here we need to use a substitution to continue:

    So our general solution to the differential equation is:

    a)

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

    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:

    First separate variables:

    Now we integrate to solve the differential equation for :

    Here we need to use a substitution to continue:

    So our general solution to the differential equation is:

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

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

    So the solutions are equivilent.

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

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

    Here we need to use a substitution to continue:

7)  Supergrowth:

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

    a)  

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

    Here we need to use some substitutions to continue: