1. Consider the ODE
ay" + by" + cy = 0
(a) Suppose that b2 - 4ac = 0. Show that e-bt/2a is a solution of the equation.
(b) Use the method of reduction of order to derive a second independent solution.
(c) Use the Wronskian to explicitly verify that you have found a fundamental set of solutions.
2. (a) Solve the IVP
ty' + 2y = sin t/t, y(1) = y0
(b) For what values of y0 does your solution approach a finite value as t → 0? What is this value?
3. Consider the equation for a mass on a spring in the absence of external forces and damping
mu" + ku = 0
(a) Let:
E(t) = 1/2m(u')2 + 1/2ku2
By considering E' (t) show that E(t) = E(0)
(b) Give a brief physical interpretation of the result in part a).
(c) Suppose we write u' = v. Describe the solution trajectories in the uv-plane.
4. Consider the differential equation
y" + 2y' = 1 + t2 + et
(a) Solve the corresponding homogeneous problem using initial conditions y(0) = 3, y'(0) = 4.
(b) Find a particular solution of the nonhomogeneous equation (it doesn't have to satisfy the initial conditions from part a)).
(c) Find the general solution to the nonhomogeneous equation.
5. (a) Compute the Laplace transform of
f(t) = teat
(b) Use the Laplace transform to solve the IVP
y" - 6y + 9y = 0, y(0) = 0, y'(0) = 2
6. (a) Classify the behavior of the system
in terms of α.
(b) Draw a phase portrait for each of the following cases: α = -1, α = 0, α = 1. Be sure to include a few sample trajectories.
7. Consider the system
(a) Find the eigenvalues and eigenvectors of the 2 * 2 matrix A given above.
(b) Diagonalize A. That is, find a matrix T and a diagonal matrix D with T-1AT = D.
(c) Making the subsitution x = Ty gives the following system for y:
y' = Dy
where T and D are as computed in the previous part. Find a fundamental matrix of this system by computing eDt
(d) Using that x = Ty, find a fundamental matrix for the original system.
8. Consider the system
(b) Find a particular solution of the nonhomogeneous system.
(c) Find the general solution of the nonhomogeneous system.