Higher Order Differential Equations Recall that the order of a differential equation is the highest derivative that appears in the equation. So far we have studied first and second order differential equations. Now we will embark on the analysis of higher order differential equations. We will restrict our attention to linear differential equations. The general linear differential equation can be written as
dny The good news is that all the results from second order linear differential equation can be extended to higher order linear differential equations. We list without proof the results Example Determine if the following functions are linearly independent y1 = e2x y2 = sin(3x) y3 = cos x Solution First take derivatives y’1 = 2e2x y’2 = 3cos(3x) y’3 = -sin x y”1 = 4e2x y”2 = 9sin(3x) y”3 = -cos x The Wronskian is Now plug in x = 0 (or any other value for x) to get (1)(-3 – 0) – (0)(-2 + 0) + (1)(0 – 12) = -15 In particular, since this is a nonzero number, we can conclude that the three functions are linearly independent. Example Use Abel’s theorem to find the Wronskian of the differential equation ty(iv) + 2 y”’ – tety” + (t3 – 4t)y’ + t2sin ty = 0 Solution We first divide by t to get y(iv) + 2/t y”’ – ety” + (t2 – 4)y’ + t sin t y = 0 Now take the integral of 2/t to get 2ln t The Wronskian is thus c e2ln t = c t2 Back to the Application and Higher Order Equations Home Page Back to the Differential Equations Home Page Back to the Math Department Home Page e-mail Questions and Suggestions |