Notes-HigherOrderLinEq

Notes-HigherOrderLinEq - Higher Order Linear Equations with...

This preview shows pages 1–3. Sign up to view the full content.

Higher Order Linear Equations with Constant Coefficients The solutions of linear differential equations with constant coefficients of the third order or higher can be found in similar ways as the solutions of second order linear equations. For an n -th order homogeneous linear equation with constant coefficients: a n y ( n ) + a n ±1 y ( n ±1) + … + a 2 y ″ + a 1 y ′ + a 0 y = 0, a n ≠ 0. It has a general solution of the form y = C 1 y 1 + C 2 y 2 + … + C n ±1 y n ±1 + C n y n where y 1 , y 2 , … , y n ±1 , y n are any n linearly independent solutions of the equation. (Thus, they form a set of fundamental solutions of the differential equation.) The linear independence of those solutions can be determined by their Wronskian, i.e., W ( y 1 , y 2 , … , y n ±1 , y n ) ≠ 0. Note 1 : In order to determine the n unknown coefficients C i , each n -th order equation requires a set of n initial conditions in an initial value problem: y ( t 0 ) = y 0 , y′ ( t 0 ) = y′ 0 , y ″( t 0 ) = y 0 , and y ( n ±1) ( t 0 ) = y ( n ±1) 0 . Note 2 : The Wronskian W ( y 1 , y 2 , … , y n ±1 , y n ) is defined to be the determinant of the following n × n matrix - - - ) 1 ( ) 1 ( 2 ) 1 ( 1 2 1 2 1 2 1 .. .. : : : " .. .. " " ' .. .. ' ' .. .. n n n n n n n y y y y y y y y y y y y .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Such a set of linearly independent solutions, and therefore, a general solution of the equation, can be found by first solving the differential equation’s characteristic equation: a n r n + a n -1 r n -1 + … + a 2 r 2 + a 1 r + a 0 = 0. This is a polynomial equation of degree n , therefore, it has n real and/or complex roots (not necessarily distinct). Those necessary n linearly independent solutions can then be found using the four rules below. (i).
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 8

Notes-HigherOrderLinEq - Higher Order Linear Equations with...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online