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Show that the first derivatives of the Legendre polynomials satisfy a self-adjoint differential equation with eigenvalue:...

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Advance Math

Show that the first derivatives of the Legendre polynomials satisfya self-adjoint differential equation with eigenvalue: lamda =n(n+1)-2

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The Legendre Differential Equation is31 1 x2 y 2xy nn 1y 0 n R x 1 1We know that x 0 is an ordinary point of equation 31 Wesee that when we divide by the coefficient 1x2 thatx 1 1 We will see later that the property of orthogonalityfalls out on the interval 1 1 by the SturmLiouville Theory Inorder to find the series solution to this differential equation wewill use the power series methodLet yx k0 to akxkyx k1 to akkxk1yx k2 to akkk 1xk2Insert these terms into the original equation 31 toobtain1 x2 k2 to akkk 1xk2 2x k1 to akkxk1 nn 1k0 to    See Answer
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