I'm not sure you're doing the second derivative properly. I agree with the first derivative, but you have to use the chain rule on the second derivative as well. See here.
Well, continue simplifying, and substitute back in so you just have x's. It's six of one, a half-dozen of the other.
Incidentally, this method of solution is inferior to the usual Cauchy-Euler method, because with this solution, you are artificially constraining x to be positive, whereas that is not the case with the usual Cauchy-Euler method.
You made two mistakes. The first one was in not noticing that the multiplies both terms in the expression The second mistake is the mysterious that multiplies the as your last term on the LHS. That seems to come out of thin air. The multiplying the term does NOT multiply the term.
Perhaps both of these mistakes are a result of not noticing the parentheses carefully enough? You absolutely have to pay attention to details like parentheses if you're going to avoid mistakes.
ok but how this method differs the y=x^a substitution method
my prof said that the exponent method is better because it solves equations that
in wikipedea they dont say anything about the difference between them
Cauchy eiler link
on what type i need to use what method
I'm not overly familiar with the differences between the two methods, other than the observation I made before. If you have a bona-fide Cauchy-Euler equation, then the x^a method should solve it. The exponential substitution might work for other types of DE's, but as I said, you're restricting yourself to positive x's. That might not matter so much if your x is really a t, and you're only interested in positive time.