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