A1 Compute the Wronskian of the functions y₁ = e¯ª, y2 =1-e* and use Theorem 1 toshow that y₁, y2 cannot be solutions of a differential equation y” + p(x)y' + q(x)y = 0with continuous functions p(x), q(x) on the interval -1 < x < 1.

The Wronskian of two functions y₁ and y₂ is defined as:W(y₁, y₂) = y₁y₂' - y₂y₁'Therefore, for y₁ =…

Answered: A1 Compute the Wronskian of the…

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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EXERCISE 1C Legendre's differential equation is given - (1-x²)y" — 2xy' + a(a + 1)y = 0, y = y(x),a ≤R, (1). Show that the general solution of (1) is y = aoy₁ (x) + a₁₂3₂ (x), where and y₁(x) = 1 a(a − 1) 2 +00 x² - Σ² n2=2 a(a + 1)/ 2n a(a +1D).... 1. 6 Y₂(x) = x + Σ₁¹ (1 — ala+¹))... (1 +00 1 1 - 2n+1 2 a(a + 1) (2n-2) (2n-1)' a(a+1) 2n(2n-1) -2n+1 --)x² -)x²n 2n
(1) A more detailed version of Theorem I says that, if the function f(r,y) is continuous near the point (a, b), then at least one solution of the differential equation y'=f(r, y) exists on some open interval I containing the point z = a and, moreover, that if in addition the partial derivative of/oy is continuous near (a, b), then this solution is unique on some (perhaps smaller) interval J. Determine whether existence of at least one solution of the given initial value problem is thereby guaranteed and, if so, whether uniqueness of that solution is guaranteed. dy (a) = 2r²y²; (b) =ln(y); dy (c) y- =H 1; dr. dr y(1) = -1. y(1) = 1. y(0) = 1.
! Find a second solution y2(r) of the differential equation 22y" + 6xy' + 6y = 0, r>0, by the method of reduction of order given that yı(x) = is a solution of the equation. Compute the Wronskian and show that y1 and y2 are linearly independent.
4. Are y1 (x) = e*and y2 (x) = x +1 solutions of the D.E. xy" – (x + 1)y' + y = x2. Are y1(x) independent? and y2(x) linearly

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A1 Compute the Wronskian of the functions y₁ = e, y₂ = 1 - e and use Theorem 1 to show that y₁, y2 cannot be solutions of a differential equation y" + p(x)y'+ q(x)y = 0 with continuous functions p(x), q(x) on the interval -1 < x < 1.