Thread: General solution of a transformed differential equation

I have a differential equation of tx''+x'+tx=0; the initial conditions are x(0)=1 and x'(0)=0.

I perform the Laplace transformation and I obtain the expression:
((s^2)+1)*X'(s)+s*X(s)=0. I have verified this with my textbook.

The problem is that the textbook then says that X'(s)/X(s) = -s/((s^2)+1) of which I understand but it suddenly jumps to conclude that
X(s) = C/((s^2)+1)^(0.5) where C is non-zero.

I have looked at a couple other examples and I see the pattern that the answer is generally X(s) = C/(coefficient of X'(s))^((coefficient of X(s))/2) if there's an 's' next to X(s) and simply C/(coefficient of X'(s))^((coefficient of X(s))).

So what is the proper way to do this?

I believe what the book did was integrate both sides



so



and solving for gives the books answer.