2(x+1)y'=y?

Question

Detertmine the radius of convergence of the resulting series (power series)

Answer 1

power series :

I assume you already know what power series is.
But to remind,

let y=SUMATION(from k=0, to k= infinity){Ck *X^k}

then y' = SUMATION(from k=0 to k=infinity){Ck+1 * X^(k-1) * K }

thus, now plug into equation

2 * ( x+1 ) * Y' = Y

then calculate C0, C1, C2 .... and find pattern

Answer 2

Well to solve the DEQ requires separation of variables.

2(x+1)y'=y

2(x+1)dy/dx=y

2dy/y=dx/(x+1)

2ln(y)=ln(x+1)

y=(x+1)^.5

x>-1

Not a power series so there may be another solution to this DEQ.

Answer 3

y´/y= 1/2 *1/(1+x)
lnIyI = 1/2 lnI1+xI +K

y= CI1+xI^1/2 This is the general solution