tailieunhanh - Ebook Advanced engineering mathematics (9th): Part 2

(BQ) Part 2 book "Advanced engineering mathematics" has contents: Complex numbers and functions, complex integration; complex integration; laurent series, residue integration; complex analysis and potential theory; numerics in general; numeric linear algebra,.and other contents. | CHAPTER J Higher Order Linear ODEs In this chapter we extend the concepts and methods of Chap. 2 for linear ODEs from order Il 2 to arbitrary order II. This will be straightforward and needs no new ideas. However the formulas become more involved the variety of roots of the characteristic equation in Sec. becomes much larger with increasing n and the Wronskian plays a more prominent role. Prerequisite Secs. . References and Answers to Problems App. 1 Part A and App. 2. Homogeneous Linear ODEs Recall from Sec. that an ODE is of nth order if the ỉth derivative y01 dny dxn of the unknown function y x is the highest occurring derivative. Thus the ODE is of the form Í . dnv Fix y y 0 y n where lower order derivatives and y itself may or may not occur. Such an ODE is called linear if it can be written 1 y n Pn-iW 1 pMy Po x y r x . For n 2 this is 1 in Sec. with Pi p and Po q . The coefficients p0 pn_1 and the function r on the right are any given functions of X and y is unknown. y n has coefficient I. This is practical. We call this the standard form. If you have pn x y n divide by Pnlx to get this form. An nth-order ODE that cannot be written in the form 1 is called nonlinear. If r x is identically zero r x 0 zero for all X considered usually in some open interval . then 1 becomes 2 y n p -i x y n n h x y Po x y 0 and is called homogeneous. If r x is not identically zero then the ODE is called nonhomogeneous. This is as in Sec. . A solution of an nth-order linear or nonlinear ODE on some open interval 7 is a function y 7i x that is defined and n times differentiable on and is such that the ODE becomes an identity if we replace the unknown function y and its derivatives by h and its corresponding derivatives. 105 106 CHAP. 3 Higher Order Linear ODEs Homogeneous Linear ODE Superposition Principle General Solution Sections will be devoted to homogeneous linear ODEs and Sec. to nonhomogeneous linear ODEs. The basic .

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