Higher Order Differential Equations
Overview
Notation
For the purpose of simplifying notation, the derivative operator is often written as
{% D = \frac{d}{dx} %}
Constraints
Homogenous and Inhomogenous Equations
An homogenous equation is one of the form
{% a_n(x) D^n y(x) + ... + a_1(x) D y(x) + a_0(x) y = 0 %}
For any two solutions to an homogenous equation, {% y_1(x) %} and {% y_2(x) %}, the sum {% y_1(x) + y_2(x) %}
is also a solution.
An Inhomogenous equation has the form.
{% a_n(x) D^n y(x) + ... + a_1(x) D y(x) + a_0(x) y = g(x) %}
For any solution of an Inhomogenous equation {% y_i(x) %}, and for any solution {% y_h(x) %} to the homogenous equation formed by setting
{% g(x) %} to 0, the sum {% y_i(x) + y_h(x) %} is also a solution.
Solution Methods
