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  3. y''−2xy' + λy = 0, −∞ < x &

y''−2xy' + λy = 0, −∞ < x < ∞ where λ is a constant,...

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Advance Math

y''−2xy' + λy = 0, −∞ < x < ∞ where λ is a constant, isknown as the Hermite equation, named after the famous mathematicianCharles Hermite. This equation is an important equation inmathematical physics.

• Find the first four terms in each of two solutions about x = 0and show that they form a fundamental set of solutions

• Observe that if λ is a nonnegative even integer, then one orthe other of the series solutions terminates and becomes apolynomial. Find the polynomial solutions for λ = 0,2,4,6,8,10.Note that each polynomial is determined only up to a multiplicativeconstant.

• The Hermite polynomial, Hn(x), is defined as the polynomialsolution of the Hermite equation with λ = 2n for which thecoefficient of xn is 2n. Find H0(x),...,H5(x).

• Give some plots of these polynomials

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