Question

Question 2 Consider the differential equation We saw in class that one solution is the Bessel function (a) Suppose we have a solution to this ODE in the form y-Σχ0CnXntr where cn 0. By considering the first term of this series show that r must satisfy r2-4-0 (and hence that r = 2 or r =-2) (b) Show that any solution of the form y-ca:0G,2n-2 must satisfy C0 (c) From the theory about singular solutions we know that a linearly inde- pendent solution is in the form y = CJ2(x) In(x) + w(x) where w(x) = bnXn-2 and b0メ0. Explain using the above parts why C 0. =0 (d) Explain why any solution of the form y-Σn=0Cnz"+r must be a scalar multiple of Jo(x) (ie a function of the form cJ2(x) for some c R) (e) Show that w must satisfy x2w" + xu1+ (x2-4)w =-2CRĂ(z) (f) Compute bk for k = 0.1, 2, 3 by equating coefficients

Answer 1

Suppose the Soluhon of the equation op 스0

Do n-ク the valuc J, y'+y" " ne o

of χ 3 3 Combinat on of 2 Hence

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