The differential equation y” = xy with initial conditions

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The differential equation y” = xy with initial conditions y(0) = 1, y'(0) = 0 looks harmless. However, it does not have a solution that can be written as an elementary function. Let’s try to solve this equation in terms of a power series y(x) = a o + a l x + a 2 z 2 + a 3 x 3 + a 4 x 4 + • • • a) Find the first 10 coefficients of the power series by writing both sides of the differential equation in terms of the power series and comparing coefficients (coefficients of corresponding powers of x have to be equal!). b) Describe the pattern underlying the coefficients. What is the general term an? c) Apply the ratio test to two consecutive non-zero terms of the power series to calculate its radius of convergence.

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