List of Notes - 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=0 ,1 ,2 ,..., 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 section about linear independence that 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)` .