3. Consider the non-homogeneous Cauchy-Euler equationax²y" + bry' + cy = f(x)(a) Let y₁(x) and y2 (r) be solutions to the associated homogeneous equation, and let y(x)be a solution to the non-homogeneous equation. Is y(x) = C₁y₁(x) + C2Y2(x) +Yp(x) thegeneral solution to the non-homogeneous ODE? Prove or provide a counterexample.(b) Solve x²y" + xy' - 9y = 5e²¹ using any method.

Answered: Image /qna-images/answer/18febcf0-9883-4fb6-9c06-dfe432d21928.jpg

Answered: 3. Consider the non-homogeneous…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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2. Suppose that y₁ is a solution for of the homogeneous SOLDE " + p(t)y' + q(t)y = 0. (a) Explain how to find a non constant function tv(t) such that y2(t) = v(t)y₁(t) is also a solution. (b) Explain why y₁ and y₂ are linearly independent.
This is the first part of a two-part problem. Let P=[-: 1 5₁(t) = [(41) 5₂(t) = - sin(4t). a. Show that y₁ (t) is a solution to the system ÿ' = Pÿ by evaluating derivatives and the matrix product y(t) = = 0 [1] -4 Enter your answers in terms of the variable t. -4 sin(4t) -4 cos(4t)] ÿ₁ (t) [181-18] b. Show that y₂ (t) is a solution to the system ÿ' = Pÿ by evaluating derivatives and the matrix product Enter your answers in terms of the variable t. 04] 32(t) = [-28]|2(t) 181-181
4. (a) find and solve the indicial equation, (b) determine the appropriate form of each of the two linearly independent solution (c) find the first five nonzero terms of the two linearly independent solutions. (i) (ii) xy" + (1-x) y' + y = 0 x(x-1) y" + 3y' -2y = 0 5. Find only the form that the two linearly independent solution should take (i) 25x(1-x²) y" - 20(5x-2)y' + (25x- 4/x)y = 0
a.) Form the complimentary solution to the homogeneous equation. y_c(t) = c_1 [ _ _] + c_2 [ _ _] b.) Construct a particular solution by assuming the form y_P(t) = e^(-3t)a and solving for the undetermined constant vector a. y_P(t) = [ _ _] c.) Form the general solution y(t) = y_c(t) + y_P(t) and impose the initial condition to obtain the solution of the initial value problem. [y_1(t) y_2(t)] = [ _ _]
Consider the following IVP: y" + 3y + 2y = 12x²; y(0) = a, y'(0) = b Find the solution. A y(x) = c₁₂e-²x + c₂e (B © 1 c₂e-2x+6x³-18x²+2 y(x) = c₁₂e-²x + c₂ex + 3x² - 18x+21 -3x y(x) = c₂₁e² + c₂e -2x + 2x² - 6x +4 y(x) = c₂e²x + c₂e² -2x+6x²-18x+21
Use the method of undetermined coefficients to determine the form of a particular solution for the given equation. y'"' + 10y" – 11y=xe* + 4x What is the form of the particular solution with undetermined coefficients? Yp(x) = ] (Do not use d, D, e, E, i, or I as arbitrary constants since these letters already have defined meanings.)
Consider the followiing equation (t^2)y′′−6y =0, t > 0, check that y1 =t^3 and find y2 and write the solution of y(t) Choose your favorite method to find the second linearly independent solution (Abel's theorem or reduction or order). Choose the correct form of the general solution from the list below.
1) Show that the given function y(x) = C,e*+C,e*cosx is a solution to the equation y"-3y" +4y'-2y 0 for any pair of constants (C, ,C) 2) Show that there is no pair of constants (C, ,C, for which y(x) above satisfies the following ICs y(0) = 1, y'(0) = 0, y"(0) = 0
2. Which of the following is a general solution to the following: x²y" + xy' + (36x² - 1) y (Hint: As discussed in the lecture, use Y, only when J, and J-, are linearly dependent). A. y = c₁J₁(2x) + C₂J_1(2x) 6 B. y = C₁J₁(x) + C₂Y₁(x) 3 3 C. y = c₁₂/₁(6x) + C₂Y₁(6x) 0 D. y = c₁J₁(6x) + c₂] _1(6x) 2
Solutions, if they exist, of equations of the form an(x)y(m) (x) + an-1(x)y(n-1)(x)+ ... + a1 (x)y'(x) + ao(x)y(x) = f(x) are completely described by n linearly independent functions. Please, stop posting the same copied solution from chegg. I need to prove this is false using the EXISTENCE AND UNIQUENESS THEOREM. I know the answer, please just explain the E&U aspect of it.

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