Problem 4: Take a deep breath! Then take two derivatives with respect to time of this expression: Q(t) = Qo exp( -, t) cos( wt + $) (1) The first derivative will have two terms, one with a cosine and one with a sine. The second derivative will have four terms, two with a cosine and two with a sine. Now consider the differential equation: d2Q , R dQ 1 dt2 L dt LC Q =0 (2) This differential equation can be satisfied by that form for Q(t), provided that y has some specific relationship to L, R and w has some specific relationship to L, R, C. What is that relationship? Hint: To find this relationship, all seven terms (one from the zeroth deriva- tive, two from the first derivative, and four from the second derivative) must add to zero. But even more importantly, because this equation must be true at all times, all the cosine terms must sum to zero and all the sine terms must separately sum to zero. Does that help? Alternative hint: If you are comfortable with complex numbers, convert the expression for Q(t) into a complex expression, and solve the problem that way. This solution uses far less paper. But it requires facility with complex numbers