9. Recall that we can define an inner product on the set of piecewise continuousfunctions PC [a, b] as follows:a.(f,g) = f*f(x)g(x)dxand we can define a norm on the same set as:||f|| = √(f, f)Legendre polynomials arise as solutions to a differential equation(1 - t²)y" — 2ty' + n(n + 1)y = 0Where an ex licit formula for the polynomials is given by:1 dnP₁ (t) =-(t²-1)"t² — and P3 (t) = t³ - ²/tb.2"n! dtnShow the Legendre polynomials P₂ (t)orthog nal on the interval [-1,1].aCompute the norm: ||P3 (t)|| on the same interval.t are

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Answered: 9. Recall that we can define an inner…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

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9. Recall that we can define an inner product on the set of piecewise continuous functions PC [a, b] as follows: a. (f,g) = b. f f(x and we can define a norm on the same set as: ||f|| = √(f, f) Legendre polynomials arise as solutions to a differential equation (1-t2)y" - 2ty' + n(n+1)y = 0 Where an explicit formula for the polynomials is given by: 1 dn Pn (t) = -(t²-1)" 2nn! dtn Show the Legendre polynomials P₂ (t): orthogonal on the interval [-1,1]. f(x)g(x) dx 3 t² and 3(t) = t³ t - Compute the norm: ||P3 (t)|| on the same interval. a t are
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