solve using power series. y'+xy=0?

Question

solve using power series

Answer 1

Let y = a(0)+a(1)x+a(2)x^2+...+a(n)x^n+...
y' = a(1)+2a(2)x+...+n*a(n)x^(n-1)+...
y'+xy = 0
a(1) + [a(0)+2a(2)]x + ... = 0
Let a(0) = 1 at first.
Comparing coefficients, we have
a(i) = 0, if i is odd.
a(0) = 1
a(2) = -1/2
...
y = 1 - x^2/2 + ...
Can you finish it now?

Answer 2

If you separate variables
dy/y = -xdx
so y=Ce^-x^2/2 which has only even powers of x in its series

Answer 3

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