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).