Question

Consider the following eigenvalue problem x 2φxx + xφx + λφ = 0, 1 < x...

Consider the following eigenvalue problem x 2φxx + xφx + λφ = 0, 1 < x < 3, φ(1) = 0, φ(3) = 0, where λ is the eigenvalue and φ(x) is the eigenfunction.

(a) Write the eigenvalue problem in standard Sturm-Liouville form (p(x)φ 0 ) 0 + q(x)φ + λσ(x)φ = 0.

(b) Show that λ ≥ 0.

(c) Use the Rayleigh quotient to obtain an upper bound for the lowest eigenvalue of the eigenvalue problem. Justify your choice of the trial function but you do not need to evaluate the integrals

Math   Advanced-Math   
0 0
Add a commentTranscribed image text
Next >

Homework Answers

Answer #1

0 0
Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Log In

Sign Up

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Use the eigenfunction expansion to solve utt = uxx + e −t sin(3x), 0 < x...
Use the eigenfunction expansion to solve utt = uxx + e −t sin(3x), 0 < x < π u(x, 0) = sin(x), ut(x, 0) = 0 u(0, t) = 1, u(π, t) = 0. Your solution should be in the form of Fourier series. Write down the formulas that determine the coefficients in the Fourier series but do not evaluate the integrals
1. y=∫upper bound is sqrt(x) lower bound is 1, cos(2t)/t^9 dt using the appropriate form of...
1. y=∫upper bound is sqrt(x) lower bound is 1, cos(2t)/t^9 dt using the appropriate form of the Fundamental Theorem of Calculus. y′ = 2. Use part I of the Fundamental Theorem of Calculus to find the derivative of F(x)=∫upper bound is 5 lower bound is x, tan(t^4)dt F′(x) = 3. If h(x)=∫upper bound is 3/x and lower bound is 2, 9arctan t dt , then h′(x)= 4. Consider the function f(x) = {x if x<1, 1/x if x is >_...
Consider the joint density function f (x, y) = 1 if 0<= x<= 1; 0<=y<= 1....
Consider the joint density function f (x, y) = 1 if 0<= x<= 1; 0<=y<= 1. [0 elsewhere] a) Obtain the probability density function of the v.a Z, where Z = X^2. b) Obtain the probability density function of v.a W, where W = X*Y^2. c) Obtain the joint density function of Z and W, that is, g (Z, W)
In this problem we consider an equation in differential form ???+???=0 The equation (6?^(1/?)+4?^3?^−3)??+(6?^2?^−2+4)??=0 in differential...
In this problem we consider an equation in differential form ???+???=0 The equation (6?^(1/?)+4?^3?^−3)??+(6?^2?^−2+4)??=0 in differential form ?˜??+?˜??=0 is not exact. Indeed, we have ?˜?−?˜?/M=______ is a function of ? alone. Namely we have μ(y)= Multiplying the original equation by the integrating factor we obtain a new equation ???+???=0 ?=____ ?=_____ Which is exact since ??=______ ??=_______ are equal. This problem is exact. Therefore an implicit general solution can be written in the form  ?(?,?)=?F where ?(?,?)=__________
obtain the green's function for the boundary value problem y''+y = f(x), y(0)=y(1)=0
obtain the green's function for the boundary value problem y''+y = f(x), y(0)=y(1)=0
A) In this problem we consider an equation in differential form ???+???=0Mdx+Ndy=0. The equation (5?^4?+4cos(2?)?^−4)??+(5?^5−3?^−4)??=0 in...
A) In this problem we consider an equation in differential form ???+???=0Mdx+Ndy=0. The equation (5?^4?+4cos(2?)?^−4)??+(5?^5−3?^−4)??=0 in differential form ?˜??+?˜??=0 is not exact. Indeed, we have My-Mx= ________ 5x^4-4(4)cos(2x)*y^(-5)-25x^4 (My-Mn)/M=_____ -4/y in function y alone. ?(?)= _____y^4 Multiplying the original equation by the integrating factor we obtain a new equation ???+???=0 where M=____ N=_____ which is exact since My=_____ Nx=______ This problem is exact. Therefore an implicit general solution can be written in the form  ?(?,?)=? where ?(?,?)=_________ Finally find the value...
consider the function f(x)= -1/x, 3, √x+2 if x<0 if 0≤x<1 if x≥1 a)Evaluate lim,→./ f(x)...
consider the function f(x)= -1/x, 3, √x+2 if x<0 if 0≤x<1 if x≥1 a)Evaluate lim,→./ f(x) and lim,→.2 f(x) b. Does lim,→. f(x) exist? Explain. c. Is f(x) continuous at x = 1? Explain.
Consider the following Linear Programming model: Maximize x+2.5y Subject to x+3y<=12 x+2y<=11 x-2y<=9 x-y>=0 x+5y<=15 x>=0...
Consider the following Linear Programming model: Maximize x+2.5y Subject to x+3y<=12 x+2y<=11 x-2y<=9 x-y>=0 x+5y<=15 x>=0 y>=0 (a) Draw the feasible region for the model, but DO NOT draw the objective function. Without graphing the objective function, find the optimal solution(s) and the optimal value. Justify your method and why the solution(s) you obtain is (are) optimal. (4 points) (b) Add the constraint “x+5y>=15” to the Linear Programming model. Is the optimal solution the same as the one in (a)?...
4.2 Given a sequence x(n) for 0 ≤ n ≤ 3, where x(0) = 4, x(1)...
4.2 Given a sequence x(n) for 0 ≤ n ≤ 3, where x(0) = 4, x(1) = 3, x(2) = 2, and x(3) = 1, evaluate its DFT X(k). 4.5 Given the DFT sequence X(k) for 0 ≤ k ≤ 3 obtained in Problem 4.2, evaluate its inverse DFT x(n).
(* Problem 3 *) (* Consider differential equations of the form a(x) + b(x)dy /dx=0 *)...
(* Problem 3 *) (* Consider differential equations of the form a(x) + b(x)dy /dx=0 *) \ (* Use mathematica to determin if they are in Exact form or not. If they are, use CountourPlot to graph the different solution curves 3.a 3x^2+y + (x+3y^2)dy /dx=0 3.b cos(x) + sin(x) dy /dx=0 3.c y e^xy+ x e^xydy/dx=0
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT