I believe what the book did was integrate both sides
so
and solving for gives the books answer.
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?