MEC702_Wk_6_Lect_2 - MEC702 Wk 6 Lect 2 Higher Order Linear...

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Unformatted text preview: MEC702 Wk 6 Lect 2 Higher Order Linear ODEs Homogneneous Linear ODEs • The ODE is of nth order if it includes the nth derivative y (n) = d4 y dx4 of the unknown functiony(x) in the Ordinary Dierential Equation. Thus the ODE is of the form F (x, y, y 0 , · · · , y (n) ) = 0 where the lower derivatives and y itself may or may not occur. Such an ODE is called linear if it can be written y (n) + pn−1 (x)y (n−1) + · · · + p1 (x)y 0 + p0 (x)y = r(x) The coecients p0 , · · · , pn−1 and the function r on the right are the given functions of x. • y and all its derivatives y (n) have the coecient 1. • All nth-order ODE that can not be written in the form above is called . non-linear • If r(x) = 0, then y (n) + pn−1 (x)y (n−1) + · · · + p1 )xy 0 + p0 (x)y = 0 is homogeneous. If r(x) ≡ 0, then the ODE is called non-homogeneous. • A solution of an nth-order (linear or linear) ODE on some open interval I is a function y = h(x) that is dened andn times dierentiable on I 1 Homogeneous Linear ODE: Superposition Principle, General Solution .The basic superposition principle (given for 2nd order homogeneous linear ODE) extends to nth order homogenous linear ODEs as follows: Theorem. (Fundamental Theorem for the Homogeneous Linear ODE) For a homogeneous linear ODE of nth order, the sums and constant multiples of soI are again solutions on I . (This does not hold lutions on some open interval for any non-homogeneous or non-linear ODE!!) Theorem. (General Solution, Basis, Particular Solution) A general solution of y (n) + pn−1 (x)y (n−1) + · · · + p1 )xy 0 + p0 (x)y = 0 on an open interval I is a solution of the ODE on I of the form y(x) = c1 y1 (x) + · · · + cn yn (x) where I; y1 , · · · , yn is a basis (or fundamental system) of solutions of the ODE on that is, these solutions are linearly independent on I. • A particular solution of the above ODE on I is obtained if we assign specic values to the n constraints c1 , . . . , cn in the solution. IVP and n Initial Conditions An initial value problem for the nth order linear homogeneous ODE consists of n initial conditions y(x0 ) = K0 , y 0 (x0 ) = K1 , . . . , y (n−1) (x0 ) = Kn−1 with given x0 , K0 , K1 , . . . , Kn−1 . 2 Extension of Methods from 2nd order ODE to nth order ODE The following are the methods we learnt in solving ODEs of 2nd order: • λ-method - solving homogeneous linear of 2nd order ODE • Euler-Cauchy Equations of 2nd order • Wronkian - testing linear independence of basis of solutions for 2nd order ODE • Method of Undetermined Coecients - solving linear non-homogeneous ODE of 2nd order • Method of Variation of Parameters - solving linear non-homogeneous ODE of 2nd order Note that each of these methods can be extended to solving ODE of nth order!!!! Example. 1. Solve the fourth-order ODE y (4) − 5y 00 + 4y = 0. 2. Solve y 000 − 2y 00 − y 0 + 2y = 0. 3. Solve the IVP y 000 − y 00 + 100y 0 − 100y = 0, 4. Solve the ODE y(0) = 4, y 0 (0) = 11, y 00 (0) = −299. y v − 3y iv + 3y 000 − y 00 = 0. 5. Solve the IVP x3 y 000 − 3x2 y 00 + 6xy 0 − 6y = 0, y(1) = 2, y 0 (1) = 1, 6. Check whether the functions y1 = e−2x , y2 = e−x , are linearly independent or dependent. 3 y3 = ex y 00 (1) = −4. ...
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