1. The oscillator has the particular solution

   

   with r and as given in section 1.9:

   

   The full solution will also have the general, underdamped homogeneous oscillator solution added to , with coefficients fixed by the initial conditions. (This general homogeneous solution is also called the transient solution, because it decays exponentially in time.) Here, however, we're asked to find initial conditions such that steady state motion begins instantly; that is, we don't have to wait for the transient to decay, because its coefficients are zero. In that case the full solution is , with velocity

   

   This obeys initial conditions

   

   using the oddness of sin. Note that our derivation of the response factor in section 1.9 (especially equations 1.120 - 1.123), lets us write out and explicitly, since

   

   giving

   

   Thus our initial conditions are

   

2. You are asked to find the particular (steady state) solution to

    

   In section 1.9 we derived the solution when the right hand side of the equation (the source) was :

    

   Note, for complex solutions z = w + ix, that the imaginary part of equation (2) is exactly equation (1), since . Thus, knowing our solution z to equation (2) gives us the solution x for a sinusoidal source, since .

   Recall that equation (2) gave the particular solution

   

   where

   

   Thus

   

3. 1. We're asked to find a particular solution to the ODE

      

      for a critically damped oscillator with . Our ODE is thus

      

      We seek a particular solution proportional to the source, . Noting that , plugging into our ODE gives

      

      Thus we have particular solution

      

   2. Now we seek a particular solution to the ODE

      

      and are encouraged to try . This guess has derivatives

      

      Plugging into our ODE gives

      

      or, noting that and terms cancel,

      

      This works if it's a time-independent equation, that is, if n=2, in which case it gives solution 2A = f. Thus we have particular solution

      

   3. We wish to use the above particular solutions to solve

      

      with initial conditions . Note that

      

      so that, by superposition, it gives a particular solution which is half the sum of particular solutions in (a) and (b):

      

      Our most general solution is given by this particular solution plus the general solution to the homogeneous problem (a critically damped oscillator with ). This is

      

      Thus we have solution

      

      with derivative

      

      At t= 0 this gives

      

      Imposing our initial conditions gives . Thus our solution obeying these initial conditions is

      

4. 1. The RC circuit obeys

      

      as the net voltage drop along the full circuit loop must be zero. This is an ODE with constant coefficients, whose solution must obey

      

      Thus the general solution is the exponentially decaying

      

      (There is only one independent solution as this is a first order ODE.)

   2. If the initial charge on the capacitor is , then and the solution is

      

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

      

      (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.) Looking for a constant particular solution , the ODE requires

      

   4. Our full solution for the inhomogeneous ODE is plus the general solution to the homogeneous equation,

      

      Applying the initial condition q(0) = 0 gives . Thus we have solution

      

      with current

      

   5. At t=0, we have q(0) = 0 and . This corresponds to no charge on the capacitor (hence no voltage drop across it), and a voltage drop across the resistor; that is, the full source voltage is dropped across the resistor. As time goes on charge builds up on the capacitor, increasing the voltage dropped across the capacitor. This means the voltage across the resistor must drop, which it does by reducing the current I. Asymptotically, as , and . In this asymptotic state, no current flows, so no voltage is dropped across the resistor and the full source voltage is dropped across the capacitor with charge .
   6. In this case, the particular solution is constant and does indeed persist. The general solution to the homogeneous equation does indeed decay exponentially; it is ``transient'' in that its effects become negligible when . So, for the RC circuit, eventually everything dies out but the particular solution , which does indeed describe an asymptotic ``steady state.''
5. 1. For the LC circuit with voltage source , Kirchoff's law for voltage drops along the circuit loop gives

      

      (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

      

      We try a solution proportional to the source,

      

      Plugging into the ODE gives

      

      solved by

      

      defining .

   2. The full solution is the particular solution above plus the general solution to the homogeneous problem

      

      Taking our independent homogeneous solutions as we obtain

      

      with derivative

      

      Setting our initial conditions gives

      

      giving solution

      

   3. Note that we may Taylor expand near as follows:

      

      

      Taking ,

      

      Thus

      

      As , all terms with positive powers of vanish, leaving only the term:

      

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