So after using the Frobenius method on $2xy''-y'-y=0$ I get one of the results for y to be a series of the form $1+\sum_{n=1}^\infty \frac{x^n}{5\cdot7\cdot9\cdot\cdot\cdot(2n+3)n!} $ but the solution according to the book I am using is $1+3\sum_{n=1}^\infty \frac{x^n}{n!(2n+3)!!} $. I don't understand where the 3 comes from. Also the double factorial indicates the factorial expands in to a product of factors that are two less than the previous one.
Thank you in advance for any help.
Okay so by definition the double factorial is expanded in to (2x+3)(2x+1) which means that for the summation to be correct for this particular problem we need to remove a factor of 3 from every term in the series. So for example for n=1 it should be $\frac{x}{5}$ and so for the denominator it evaluates to $1!(5)(3)=~15$ which corresponds to $n!(2n+3)(2n-1)=~15$ so in order for it be correct a factor of 3 must be removed, as aforementioned.