Initial Value, Boundary Value, and Eigenvalue Problems

In a Nut Shell: There are three important types of problems involving linear,

second order, ordinary, homogeneous d.e.’s of the form

y’’ + p(x)y’ + q(x)y = 0

that frequently appear in a first course in differential equations. They include:

Initial Value Problems

Boundary Value Problems

Eigenvalue Problems.

Strategy: Start by identifying the type of problem.

Type 1: An initial value problem has conditions: y(a) = A, y’(a) = B

You saw this type of problem in studying free vibrations of a mechanical system.

my’’ + c y’ + k y = 0 along with conditions:

y(0) = y_o, dy(0)/dt = v_o (initial displacement and initial speed of system)

where: m = mass, c = damping coefficient, and k = spring rate

The procedure for solution is to find the complementary solution subject to the

initial conditions.

Type 2: A Boundary Value problem has conditions: y(a) = A, y(b) = B

Note: The boundary value problem also goes under the name of an end point problem.

Other possible boundary value conditions (or endpoint conditions) include:

y’(a) = A, y(b) = B, or y’(a) = A, y’(b) = B, or any linear combination

The procedure for solution is to find the complementary solution subject to the

end conditions.

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