4. Higher Order Linear DEs

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1 4. Higher Order Linear DEs Department of Mathematics & Statistics ASU

2 Outline of Chapter 4 1 General Theory of nth Order Linear Equations 2 Homogeneous Equations with Constant Coecients 3 The Method of Undetermined Coecients 4 The Method of Variation of Parameters The mathematical methods to solve rst and second order linear DEs will extend directly to linear DEs of higher order.

3 General Theory of nth Order Linear Equations A nth order linear DE has the following form P 0 (t) d n y dt n + P 1(t) d n 1 y dt n P n 1(t) dy dt + P n(t)y = G(t), where P 0, P 1,,P n,g C(I ) is a given function and P 0 (t) 0. Dividing the DE (1) by P 0 (t), we can have the standard form L[y] = d n y dt n (1) + p 1(t) d n 1 y dt n p n 1(t) dy dt + p n(t)y = g(t). (2) L is the linear dierential operator of order n dened by (2). In order to consider a unique solution of the IVP, we require n initial conditions; y (t 0 ) = y 0 0, y (t 0 ) = y 1 0,,y (n 1) (t 0 ) = y n 1 0. (3)

4 Theorems Theorem If p 1,p 2,,p n C(I ) with the open interval I, there is exactly one solution y = φ(t) of the DE (2) that satises the initial data (3). This solution y = φ(t) exits throughout the small open interval I t 0. The Wronskian for higher DEs is dened by y 1 y 2... y n y (1) W (y 1, y 2,,y n )(t) = 1 y (1) 2... y (1) n... y (n 1) 1 y (n 1) 2... y (n 1) n, where y 1,y 2,,y n t I. are solutions of the higher linear DE for all

5 Denitions A set of solutions y 1,y 2, y n of the Eq. L[y] = 0 whose Wronskian W (y 1,y 2,,y n ) 0 is a fundamental set of solutions. The general solution of the Eq. L[y] = 0 is expressed by a linear combination of a fundamental set of solution. Linear independence: The functions f 1, f 2,,f n are said to be linearly independent on the interval I if there constants = 0 for all i = 1,2,,n such that a i a 1 f 1 (t) + a 2 f 2 (t) + + a n f n (t) = 0 for all t I. The functions f 1, f 2,,f n are said to be linearly dependent on I if they are not linearly independent.

6 Theorems Theorem (I) If p 1,p 2,,p n C(I ) with the open interval I and y 1,y 2,,y n are solutions of the homogeneous Eq. L[y] = 0 and W (y 1,y 2,,y n )(t) 0 for at least one point t I, then every solution of the homogeneous Eq. can be expressed by a linear combination of the solutions y 1,y 2,,y n. (II) If {y 1 (t),y 2 (t),,y n (t)} is a fundamental set of solutions of L[y] = 0 on an interval I t, then y 1 (t),y 2 (t),,y n (t) are linearly independent solutions of L[y] = 0 on I. Conversely, if y 1 (t),y 2 (t),,y n (t) are linearly independent solutions of L[y] = 0 on I, then they form a fundamental set of solutions on I. According the Theorem (II), we can see that y 1 (t),y 2 (t),,y n (t) are linearly independent solutions of L[y] = 0 on I W (y 1,y 2,,y n )(t) 0 for all t I.

7 Any solution of the nonhomogeneous Eq. L[y] = g(t) can be written as y = c 1 y 1 (t) + c 2 y 2 (t) + + c n y n (t) + Y (t), where Y (t) is some particular solution of L[y] = g(t). Generally, the converse part of the Theorem (II) is not true, that is, even if W (f 1,f 2,,f n )(t) = 0 for some t I, f 1,f 2,,f n may be linearly independent. Note that we do not assume that f 1,f 2,,f n are solutions of the higher order DE L[y] = 0. We will see this in the next example #2.

8 Examples Examples 1. Determine the given set of functions is linearly dependent or linearly independent in the whole domain: (1) f 1 (t) = 2t 3, f 2 (t) = t 2 + 1, f 3 (t) = 2t 2 t. (2) f 1 (t) = 1, f 2 (t) = t, f 3 (t) = e t, f 4 (t) = te t. 2. (1) Show that f (t) = t 2 t and g(t) = t 3 are linearly dependent on the interval ( 1, 0) and (0, 1). (2) Show that f and g are linearly independent on the interval ( 1,1). (3) W (f, g)(t) = 0 for all t ( 1,1).

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