# List of Notes - Category: Nth-Order Linear ODE with Constant Coefficients

## Method of Solution

To find a solution of linear homogeneous differential equation wiith constant coefficients y^((n))+a_(n-1)y^((n-1))+...+a_2y''+a_1y'+a_0=0 we assume that solution has form y(x)=e^(rx) .

Plugging this slution into equation and noting that y^((n))=r^n e^(rx) gives

## The Method of Undetermined Coefficients

The general solution to the linear differential equation L(y)=phi(x) 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(y) = 0. For method for obtaining y_h when the differential equation has constant coefficients see method of solutions note. Here one method will be given for obtaining a particular solution y_p once y_h is known.

## Variation of Parameters

Variation of parameters, like method of undetermined coefficients, is another method for finding a particular solution of the nth-order linear differential equation L(y)=phi(x) once the solution of the associated homogeneous equation L(y) = 0 is known.

## Initial Value Problems

Itnitial-value problems are solved by applying initial conditions to the general solution of the differential equation. Note that 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 that must 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.