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y'+2xy=0

Find:

Write sereis as an elementary function

My solution so far:

y=[Sum n=0, to infinity]C(sub-n)*x^n

y'=[Sum n=1, to infinity]n*C(sub-n)*x^(n-1)

y' can be transformed into:

=[Sum n=0, to infinity](n+1)*C(sub-n+1)*x^n

([Sum n=0, to infinity](n+1)*C(sub-n+1)*x^n) + 2x([Sum n=0, to infinity]C(sub-n)*x^n)=0

([Sum n=0, to infinity](n+1)*C(sub-n+1)*x^n) +2([Sum n=0, to infinity]C(sub-n)*x^(n+1))=0

My question:

Can I transform [Sum n=0, to infinity]C(sub-n)*x^(n+1) into ([Sum n=1, to infinity]C(sub-n-1)*x^(n)?

If so, then y(x)=([Sum n=,0. to infinity]((-1)^(n+1)*C(sub-o)*x^(n+1)*2^(n-1))/n!

How do I transform that into an elementary function?

Sorry about the ugly typing...