2.II.6 B2 . \mathrm{II} . 6 \mathrm{~B} \quad

Differential Equations | Part IA, 2005

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

W+2xW=EW,-W^{\prime \prime}+2 x W^{\prime}=E W,

for EE a real constant.

Impose the condition W(0)=0W(0)=0 and determine those values of EE for which your power series gives polynomial solutions (i.e., an=0a_{n}=0 for nn sufficiently large). Give the values of EE for which the corresponding polynomials have degree less than 6 , and compute these polynomials.

Hence, or otherwise, find a polynomial solution of

W+2xW=x43x3+415x5,-W^{\prime \prime}+2 x W^{\prime}=x-\frac{4}{3} x^{3}+\frac{4}{15} x^{5},

satisfying W(0)=0W(0)=0.

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