Category: Nth-Order Linear ODE

Basic Concepts

An nth-order linear differential equation has the form `b_n(x)y^((n))+b_(n-1)(x)y^((n-1))+...+b_2(x)y''+b_1(x)y'+b_0(x)y=g(x)`, where `g(x)` and all coefficients `b_j(x)` (`j=bar(0..n)`) depend solely on the variable `x`. In other words, they do not depend on `y` or on any derivative of `y`.

Linear Independence and Wronskian

A set of functions {`y_1(x)`, `y_2(x)`, ..., `y_n(x)` } is linearly dependent on `a<=x<=b`, if there exist constants `c_1`, `c_2`, ... , `c_n`, not all zero, such that `c_1 y_1(x)+c_2 y_2(x)+...+c_n y_n(x)-=0` on `a<=x<=b`.

Particular Solution

Consider the nonhomogeneous differential equation `y^((n))+a_(n-1)(x)y^((n-1))+...+a_1(x)y'+a_0(x)y=g(x)`.

Recall from the section about linear independence that the solution of the corresponding homogeneous equation `y^((n))+a_(n-1)(x)y^((n-1))+...+a_1(x)y'+a_0(x)y=0` is given as `y_h=c_1y_1(x)+c_2y_2(x)+...+c_ny_n(x)`.