Chap2new - Sem 1 Sesi 06/07 CHAPTER 2 1 The General Theory...

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Sem 1 Sesi 06/07 18 CHAPTER 2 1 The General Theory A linear ODE of order 2 has the form ( ) ( ) ( ) y p x y q x y r x ′′ + + = (2.1) where p(x) , q(x) and r(x) are functions of x. The functions p(x) and q(x) are called the coefficients of the equation. If r(x) = 0 then (2.1) has the form ( ) ( ) 0 y p x y q x y ′′ + + = (2.2) and is then called homogeneous to indicate that all terms are of the first degree in y and its derivatives. The original equation (2.1) is called nonhomogeneous because it contains a term r(x) which does not depend on y. ( Note, however, that this use of the term homogeneous has a different meaning for first order differential equations) Example 2.1 (Ref: page 30) The equation (1 ) 3 6 0 x y xy y ′′ - - + = is a homogeneous 2 nd order linear differential equation. While the equation 3 9 cos x y y y e x ′′ + + = is a nonhomogeneous 2 nd order linear differential equation. This chapter will only discuss two cases: Linear differential equation with constant coefficients ( ) ay by cy r x ′′ + + = (2.3) The Euler-Cauchy equation 2 ( ) x y axy by r x ′′ + + = (2.4) Those with other variable coefficients will be considered in Chapter 3. Theorem 2.1 ( Principle of Superposition) If 1 ( ) y y x = is a solution of (2.2) then 1 1 ( ) y c y x = , where 1 c is an arbitrary constant is also a solution. If 1 ( ) y x and 2 ( ) y x are solutions of (2.2) then the linear combination 1 1 2 2 ( ) ( ) y c y x c y x = + (2.5) where 1 c and 2 c are arbitrary constants, is also a solution. Note: The theorem is not applicable to nonhomogeneous or nonlinear equation. See examples in Ref. pages 31 and 32. )
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Sem 1 Sesi 06/07 19 Example 2.2 (Ref: page 31) Verify that 1 2 sin cos y c x c x = + is a solution of the homogeneous equation 0 y y ′′+ = . where 1 c and 2 c are arbitrary constants. Let 1 sin y x = . Then 1 1 sin y x y ′′= - = - 1 1 0 y y ′′+ = . Similarly let 2 cos y x = . Then 2 2 cos y x y ′′ = - = - 2 2 0 y y ′′ + = . 1 sin y x = and 2 cos y x = are both solution of the equation. Hence by Theorem 2.1 their linear combination 1 2 sin cos y c x c x = + is also a solution. Definition 2.2 Two functions 1 ( ) y x and 2 ( ) y x are said to be linearly independent if the equality 1 1 2 2 ( ) ( ) 0 k y x k y x + = (2.6) holds only when 1 2 0 k k = = , i.e. when 1 ( ) y x or 2 ( ) y x are not proportional to the other. Otherwise, 1 ( ) y x and 2 ( ) y x are linearly dependent if (2.6) holds for some constants 1 k and 2 k , not both zero. Example 2.3 (Ref: page 33) Determine whether the following functions 1 ( ) y x and 2 ( ) y x are linearly dependent or independent: (a) 2 1 2 ( ) , ( ) 1 x y x e y x x = = + (b) 1 2 ( ) sin 2 , ( ) sin cos y x x y x x x = = (a) Let 2 1 2 ( 1) 0 x k e k x + + = (1) Differentiate wrt x, 2 1 2 2 0 x k e k + = (2) Again differentiate wrt x, 2 1 4 0 x k e = . (3) But for all real x , 2 0 x e . 1 0 k = . Substitute into (2)
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Chap2new - Sem 1 Sesi 06/07 CHAPTER 2 1 The General Theory...

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