Category: Nth-Order Linear ODE

Basic Concepts

An nth-order linear differential equation has the form bn(x)y(n)+bn1(x)y(n1)++b2(x)y+b1(x)y+b0(x)y=g(x){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(x){g{{\left({x}\right)}}} and all coefficients bj(x),j=0..n{b}_{{j}}{\left({x}\right)},j={\overline{{{0}..{n}}}} depend solely on the variable x{x}. In other words, they do not depend on y{y} or on any derivative of y{y}.

Linear Independence and Wronskian

A set of functions {y1(x),y2(x),,yn(x)}{\left\{{y}_{{1}}{\left({x}\right)},{y}_{{2}}{\left({x}\right)},\ldots,{y}_{{n}}{\left({x}\right)}\right\}} is linearly dependent on axb{a}\le{x}\le{b}, if there exist constants c1{c}_{{1}}, c2{c}_{{2}}, ... , cn{c}_{{n}}, not all zero, such that c1y1(x)+c2y2(x)++cnyn(x)0{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 axb{a}\le{x}\le{b}.

Particular Solution

Consider the nonhomogeneous differential equation y(n)+an1(x)y(n1)++a1(x)y+a0(x)y=g(x){{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)}}}.