Solving Constant Coefficient, Linear, Second Order, Homogeneous Equations

Constant coefficient, linear second order, homogeneous equations take the form

This is a constant coefficient equation becauseandare all constants.

It is linear because we can writeaswhereis a linear operator because

The equation is second order because of the expressionand homogeneous because there is no function ofon the right hand side.

We can solve equations of this form by assuming a solution of the formso thatandSubstitution into the equation above gives

We can divide by the non zero factorto give

This equation may have distinct real solutionsandso that

The equation may have one real solutionIn this case

The equation may have complex solutionsandIn this case

Each solution has arbitrary constantsandThese constants may be found given suitable boundary conditions (if the derivatives are with respect tothen the boundary conditions are called initial conditions).