Paper 2, Section II, 6C6 \mathrm{C}

Differential Equations | Part IA, 2019

Find all power series solutions of the form y=n=0anxny=\sum_{n=0}^{\infty} a_{n} x^{n} to the equation

(1x2)yxy+λ2y=0\left(1-x^{2}\right) y^{\prime \prime}-x y^{\prime}+\lambda^{2} y=0

for λ\lambda a real constant. [It is sufficient to give a recurrence relationship between coefficients.]

Impose the condition y(0)=0y^{\prime}(0)=0 and determine those values of λ\lambda for which your power series gives polynomial solutions (i.e., an=0a_{n}=0 for nn sufficiently large). Give the values of λ\lambda for which the corresponding polynomials have degree less than 6 , and compute these polynomials. Hence, or otherwise, find a polynomial solution of

(1x2)yxy+y=8x43\left(1-x^{2}\right) y^{\prime \prime}-x y^{\prime}+y=8 x^{4}-3

satisfying y(0)=0y^{\prime}(0)=0.

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