MATH
Chapter 3 Lecture 9

# Chapter 3 Lecture 9 - McGill University Math 263...

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McGill University Math 263: Differential Equations for Engineers CHAPTER 3: N-TH ORDER DIFFERENTIAL EQUATIONS (IV) 1 Solutions for Equations with Variable Coefficients In this lecture we will give a few techniques for solving cer- tain linear differential equations with non-constant coefficients. We will mainly restrict our attention to second order equations. However, the techniques can be extended to higher order equa- tions. The general second order linear DE is p 0 ( x ) y ′′ + p 1 ( x ) y + p 2 ( x ) y = q ( x ) . This equation is called a non-constant coefficient equation if at least one of the functions p i is not a constant function. 1.1 Euler Equations An important example of a non-constant linear DE is Euler’s equation x 2 y ′′ + axy + by = 0 , (1) 0-0

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where a, b are constants. This equation has singularity at x = 0 . The fundamental theorem of existence and uniqueness of solution holds in the region x > 0 and x < 0 , respectively. So one must solve the problem in the region x > 0 , or x < 0 separately. 1.2 The method with differential operators We first consider the region x > 0 . This Euler equation can be transformed into a constant coefficient DE by the change of independent variable. Let x = e t , and y ( x ) = y (e t ) = ˜ y ( t ) . Then we derive y d t = d y d x d x d t = e t d y d x . In operator form, we have d d x = e t D, x d d x = D, here we have set D = d d t . From the above, we derive d 2 d x 2 = e t De t D = e 2 t e t De t D = e 2 t ( D 1) D, 0-1
so that x 2 y ′′ = D ( D 1)˜ y . By induction one easily proves that d n d x n = e nt D ( D 1) · · · ( D n + 1) so that, x n y ( n ) = D ( D 1) · · · ( D n + 1)(˜ y ) . With the variable t , Euler’s equation becomes D ( D 1)˜ y + aD ˜ y + b ˜ y = d 2 ˜ y d t 2 + ( a 1) y d t + b ˜ y = q ( e t ) = ˜ q ( t ) , which is a linear constant coefficient DE. Solving this for ˜ y as a function of t and then making the change of variable t = ln( x ) , we obtain the solution of Euler’s equation for y as a function of x .

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