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