1. As noted in the preceding hanging cable problems, gravity acts unopposed on the hanging length x, with mass mx/l, giving downward force F = mgx/l (acting to increase the hanging length x). Newton II thus gives

   

   This is a differential equation with constant coefficients, with solutions of the form . Plugging this form into the ODE, and . The ODE becomes an algebraic equation for :

   

   giving the 2 linearly independent solutions . The differential equation thus has the most general solution

   

   with corresponding velocity

   

   Applying the initial conditions , gives

   

   solved by . Thus

   

   in agreement with problem set 1, problem 3d.

2. 1. Newton II for the resistive force gives

      

      As we exploited in our direct integration techniques, we can relate time derivatives of v to x-derivatives of v by using the chain rule, for v = v(x):

      

      Thus Newton II gives us an ODE for v(x):

      

   2. This is a differential equation with constant coefficients, again with solution of the form . Plugging this form into the ODE, so the ODE becomes an algebraic equation for :

      

      giving the single linearly independent solution . (There is only one solution because the ODE is first order; that is, it has only 1 derivative.) The most general solution is thus

      

      Initial conditions imply that . Thus in agreement with problem set 1, review problem 2b.

3. As in review problem 3, we have differential equation

   

   whose solution is critically damped, with general form

   

   The angular velocity is again the time derivative

   

   At t=0 this gives

   

   We now use our altered initial conditions to identify the altered coefficients for this general solution (the general solution is the same since the physical system/ODE is the same.) Equating the above to our initial conditions gives . Thus we have

   

   

4. Here our altered resistive force gives Newton II as

   

   or a damped oscillator with , so that , real and nonzero so that the motion is overdamped. Our linearly independent solutions are thus

   

   This gives the general solution

   

   with derivative

   

   At t=0 this gives

   

   Equating to our initial conditions gives A+ B = 0, . Thus , giving solution

   

   which can be rewritten

   

   is messy; it is

   

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