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a) Use the method of Lagrange to find the general solution to XZx+2xzZy = 2. In the following cases determine whether a solution exists, and if so give it's most general form. (i) z = 2x on y = 2x on y = 2x² + 1. (ii) 2= 2x² on y = 3x³. (iii) z = x² on y = x³ – 1.

a) Use the method of Lagrange to find the general solution to XZx+2xzZy = 2. In the following cases determine whether a solution exists, and if so give it's most general form. (i) z = 2x on y = 2x on y = 2x² + 1. (ii) 2= 2x² on y = 3x³. (iii) z = x² on y = x³ – 1.

Advanced Engineering Mathematics
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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3. (a) (b) Use the method of Lagrange to find the general solution to xZx + 2xzzy = 2. In the following cases determine whether a solution exists, and if so give it's most general form. (i) z = 2x on y = 2x² + 1. (ii) z = 2x² on y = 3x³. (iii) z = x² on y = x³ – 1. (i) The ends of a finite string of length are fixed. The initial displace- ment is given by the function (x) = 2 sin(4x), the initial velocity is zero. Find the vibrations of the string for t > 0,0 ≤ x ≤ π, that is, solve = a²uxx, u(t,0) = u(t, π) = 0, u(0, x) = 2 sin(4x), ut(0, x) = 0. Utta² (ii) Now assume the string is infinite with the same initial conditions. Solve the problem using d'Alembert's formula.
Transcribed Image Text:3. (a) (b) Use the method of Lagrange to find the general solution to xZx + 2xzzy = 2. In the following cases determine whether a solution exists, and if so give it's most general form. (i) z = 2x on y = 2x² + 1. (ii) z = 2x² on y = 3x³. (iii) z = x² on y = x³ – 1. (i) The ends of a finite string of length are fixed. The initial displace- ment is given by the function (x) = 2 sin(4x), the initial velocity is zero. Find the vibrations of the string for t > 0,0 ≤ x ≤ π, that is, solve = a²uxx, u(t,0) = u(t, π) = 0, u(0, x) = 2 sin(4x), ut(0, x) = 0. Utta² (ii) Now assume the string is infinite with the same initial conditions. Solve the problem using d'Alembert's formula.
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