1. We wish to solve with initial conditions .

   First we seek a particular solution for our driving force. Substituting the form , we find the differential equation is true if

   

   Thus we have the particular solution

   

   To get the most general solution possible we must add the general solution to the homogeneous problem, giving
   

   with velocity

   

   Imposing the boundary conditions at t=0 then gives

   

   which yields, after algebra,
   

   

2. We now seek the solution for a sawtooth driving force,

   

   For a specific n, defining , we obtain the particular solution with form given by section 1.9:

   

   where

   

   The full solution for the sawtooth driving force is then given by superposition,

   

3. 1. The LC circuit obeys (from Kirchoff's laws, setting the net voltage change for the full circuit loop to zero)

      

      This is just an undamped oscillator equation, with solution

      

      for . (Our usual solution method yields , or , to give the oscillating solutions we've chosen to write as sin and cos here.)

   2. We have

      

      giving initial conditions

      

      Here initial conditions are , giving , so our solution is

      

   3. With a constant voltage source in the circuit, Kirchoff's law for voltage drops across the current loop gives

      

      (Note that, traversing the circuit clockwise from the switch, we have a voltage gain followed by voltage drops and q/C; the total gain must balance the total drop.) The particular solution is just a constant , obeying

      

   4. The full solution is the sum of the particular solution (c) and the homogeneous general solution (a), with coefficients fixed by initial conditions. Thus

      

      giving initial conditions

      

      Here initial conditions are , giving , so our solution is

      

   5. Here the homogeneous solution, , does not decay in time. It persists and remains comparable to the particular solution forever. Thus the homogeneous solution is not transient, nor does the system ever reach a steady state described only by the particular solution.

4. For the RC circuit with voltage source , Kirchoff's law for voltage drops along the circuit loop gives

   

   (As in problem 3, in traversing the circuit clockwise from the switch, we have a voltage gain followed by voltage drops and q/C, with the total gain balancing the total drop.)

   We seek a particular solution to the inhomogeneous ODE

   

   As in section 1.9, it is simpler to look at the complex ODE

   

   whose real part reproduces our original ODE. When we obtain our final answer q to this complex ODE, its real part will solve our original equation.

   We write our trial solution as proportional to the source,

   

   where is time-independent. Plugging this form into our ODE gives

   

   or

   

   Writing as , in order to separate out phase and amplitude information on the response, gives

   

   Thus

   

   giving particular solution

   

   with as given above. Taking the real part to find the solution to our original real ODE,

   

   again with as given above.

   Note that the amplitude to our response is maximal when the denominator in its prefactor above is minimal (that is, when is minimal). This clearly occurs when . The magnification factor, relating the intensity of the response at to its maximal value at is given by

   

   The phase lag has tangent ranging from 0 at , through positive values to as ; thus it ranges from 0 at to as . Both the magnification factor and phase lag are plotted below.

   

• About this document ...