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
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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.