2014hw2advmath

BME 230 A

Home Work Problem Set 2 Please read Chapter 2 and 3 Due Wednesday, October 22, 2013

Applied Engineering Mathematics I

If you have forgot linear algebra, please read chapter 7 and 8, which is the pre-requisite topics that you should familiar with. 1. 2. 3. 4. 5. 6. Problem 5 on page 111 Problem 6 on page 111 Problem 11 on page 111 Problem 12 on page 111 Problem 15, on page 111 Show that each of the functions y1=3x2-x and y2=x2-x is a solution of the differential equation: x2y’’-2y=2x Determine whether y=c1y1+c2y2 is a solution for all choice of the constants c1,c2 7. For the 3rd order linear homogeneous differential equation, d3y d2y dy P3 (x) 3 + P2 (x) 2 + P1 (x) + P0 (x)y = 0 dx dx dx (i)

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Derive the differential equation for the Wronskian determinant Please show that

x P (ξ ) −∫ 2 dξ x 0 P3 (ξ )

(ii)

W ( x) = W ( x 0 )e

(iii)

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For the following third order linear homogeneous equations, find Wronskian determinant W(x).

x

d 3y d 2y dy + 2x 2 2 + sin( x) + cos( x) y = 0 3 dx dx dx

8.

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Briefly answer the following questions: • What is the superposition principle? Does it hold for nonlinear equations? For nonhomogeneous equation? For homogeneous linear equations? 1 Total page 2

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How many arbitrary constants does a general solution of a nth order nonhomogeneous linear equation involve? A general solution of a homogeneous linear equation? How many initial conditions does one need to determine all the constants?

9.

Let {u1(x), u2(x), ………un(x)} be a set of linear independent functions on interval I, (i) Prove that for

∑ a u = ∑b u

k k k =1 k =1

n

n

k k

to hold on I, it is sufficient and necessary (when and only when) that ak=bk for each k=1,…..n. (ii) 10. 11. 12. If {u1(x), u2(x), ………un(x)} are linear dependent, can you make the same conclusion? Please give an example with n=3.

Problem 13 on page 59 Problem 14 on page 59 Problem 23 on page 59

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