Category: Nth-Order Linear ODE with Constant Coefficients

Method of Solution

To find the solution of the linear homogeneous differential equation wiith constant coefficients $$${{y}}^{{{\left({n}\right)}}}+{a}_{{{n}-{1}}}{{y}}^{{{\left({n}-{1}\right)}}}+\ldots+{a}_{{2}}{y}''+{a}_{{1}}{y}'+{a}_{{0}}={0}$$$, we assume that the solution has the form $$${y}{\left({x}\right)}={{e}}^{{{r}{x}}}$$$.

Method of Undetermined Coefficients

The general solution to the linear differential equation $$${L}{\left({y}\right)}=\phi{\left({x}\right)}$$$ is given as $$${y}={y}_{{h}}+{y}_{{p}}$$$, where $$${y}_{{p}}$$$ denotes one solution to the differential equation and $$${y}_{{h}}$$$ is the general solution to the associated homogeneous equation $$${L}{\left({y}\right)}={0}$$$. For the method to obtain $$${y}_{{h}}$$$ when the differential equation has constant coefficients, see the method of solutions note. Here, one method will be given for obtaining the particular solution $$${y}_{{p}}$$$ once $$${y}_{{h}}$$$ is known.

Variation of Parameters

Variation of parameters, like the method of undetermined coefficients, is another method for finding the particular solution of an nth-order linear differential equation $$${L}{\left({y}\right)}=\phi{\left({x}\right)}$$$ once the solution to the associated homogeneous equation $$${L}{\left({y}\right)}={0}$$$ is known.

Initial Value Problems

Itnitial value problems are solved by applying the initial conditions to the general solution of a differential equation. Note that the initial conditions are applied only to the general solution and not to the homogeneous solution $$${y}_{{h}}$$$, even though it is $$${y}_{{h}}$$$ that possesses all the arbitrary constants to be evaluated. The only exception is when the general solution is the homogeneous solution, that is when the differential equation under consideration is itself homogeneous.