Higher Order Differential Equations

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
dn-1y
dy

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

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