MEC702_Wk_6_Lect_2

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

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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 Dierential 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 coecients 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 coecient 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 dened andn times dierentiable 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 specic 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 Coecients - 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. ...
View Full Document

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern