Category: Nth-Order Linear ODE

Basic Concepts

An nth-order linear differential equation has the form $$${b}_{{n}}{\left({x}\right)}{{y}}^{{{\left({n}\right)}}}+{b}_{{{n}-{1}}}{\left({x}\right)}{{y}}^{{{\left({n}-{1}\right)}}}+\ldots+{b}_{{2}}{\left({x}\right)}{y}''+{b}_{{1}}{\left({x}\right)}{y}'+{b}_{{0}}{\left({x}\right)}{y}={g{{\left({x}\right)}}}$$$, where $$${g{{\left({x}\right)}}}$$$ and all coefficients $$${b}_{{j}}{\left({x}\right)},j={\overline{{{0}..{n}}}}$$$ 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 $$${\left\{{y}_{{1}}{\left({x}\right)},{y}_{{2}}{\left({x}\right)},\ldots,{y}_{{n}}{\left({x}\right)}\right\}}$$$ is linearly dependent on $$${a}\le{x}\le{b}$$$, if there exist constants $$${c}_{{1}}$$$, $$${c}_{{2}}$$$, ... , $$${c}_{{n}}$$$, not all zero, such that $$${c}_{{1}}{y}_{{1}}{\left({x}\right)}+{c}_{{2}}{y}_{{2}}{\left({x}\right)}+\ldots+{c}_{{n}}{y}_{{n}}{\left({x}\right)}\equiv{0}$$$ on $$${a}\le{x}\le{b}$$$.

Particular Solution

Consider the nonhomogeneous differential equation $$${{y}}^{{{\left({n}\right)}}}+{a}_{{{n}-{1}}}{\left({x}\right)}{{y}}^{{{\left({n}-{1}\right)}}}+\ldots+{a}_{{1}}{\left({x}\right)}{y}'+{a}_{{0}}{\left({x}\right)}{y}={g{{\left({x}\right)}}}$$$.