1.

Let S be the solution set of (4.4).

1. Show that S is a subspace of C ²( R), the set of twice-continuously differentiable function defined on R.

2. Use the results of this section to show S is two-dimensional for any values of a, b, and c, provided only that a ? 0.

2.

Suppose (4.4) has characteristic roots ? ? i, where ?, ? ? R and ? ? 0. Show that, for any k ₁, k ₂, there is a unique choice of c ₁, c ₂ such that the solution of (4.6) is

3.

Suppose (4.4) has the single characteristic root r = ? b/(2 a). Show that, for any k ₁, k ₂, there is a unique choice of c ₁, c ₂ such that the solution of (4.6) is

4.

For each of the following IVPs, find the general solution of the ODE and use it to solve the IVP:

5.

The following differential equations are accompanied by boundary conditions auxiliary conditions that refer to the boundary of a spatial domain rather than to an initial time. By using the general solution of the ODE, determine whether a nonzero solution to the boundary value problem (BVP) exists, and if so, whether the solution is unique.

6.

Determine the values of ? ? R such...