How to solve boundary value problems with distinct real roots

We already know how to solve an initial value problem for a second-order homogeneous differential equation. Boundary value problems are very similar, but differ in a few important ways: 1) Initial value problems will always have a solution; boundary value problems may not, 2) The initial conditions given in an initial value problem relate to the general solution and its derivative; the initial conditions in a boundary value problem both relate to the general solution, not its derivative, and 3) The initial conditions given in an initial value problem are both for values of x0=0; the initial conditions given in a boundary value problem are for x0=a and x0=b.

Homogeneous differential equations initial value problems

We’ve already learned how to find the complementary solution of a second-order homogeneous differential equation, whether we have distinct real roots, equal real roots, or complex conjugate roots. Now we want to find the particular solution by using a set of initial conditions, along with the complementary solution, in order to find the particular solution.

Solving second-order homogeneous differential equations

The first thing we want to learn about second-order homogeneous differential equations is how to find their general solutions. The formula we’ll use for the general solution will depend on the kinds of roots we find for the differential equation.

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