## Problems

In Problems 1-8, find the general solution of the given differential equation.

In Problems 9-16, find the particular solution of the initial value problem.

In Problems 17-20, solve the initial value problem and find the amplitude, frequency, and phase shift of the solution.

21            A mass-spring system mx" + bx' + kx = 0 has spring constant k = 29, damping constant b = 4, and mass m = 1. At time t = 0, the position is x(0) = 2 and the velocity is x'(0) = 1. Find the position x(t) as a function of time.

22            A mass-spring system mx" + bx' + kx = 0 has spring constant k = 24, damping constant b = 12, and mass m = 3. At time t = 0, the position is x(0) = 0 and the velocity is x'(0) = -1. Find the position x(t) as a function of time.

23            Show that if y(t) is a solution of a differential equation ay" + by' + cy = 0, such that y(t0) = 0 and y'(t0) = 0 at some time t0, then y(t) = 0 for all t.

24            In the differential equation ay" + by' + cy = 0, suppose that a is positive and c is negative. Show that the characteristic equation has one positive real root and one negative real root, so that the general solution has the form y = Aert + Best where r is positive and s is negative.

25            In the differential equation ay" + by' + cy - 0, suppose that a and c are positive and b is negative. Show that there are three cases for the general solution, depending on the sign of the discriminant d:

 Case 1 If d is positive, the general solution has the form y = Aert + Best where r and s are positive. Case 2 If d is zero, the general solution has the form y = Aert + Btert where r is positive. Case 3 If d is negative, the general solution has the form y = eαt[A cos (βt) + B sin (βt)] where α is positive, so that the graph is an oscillation whose amplitude is increasing instead of decreasing.

Last Update: 2006-11-25