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).