In my lectures notes for solving ODEs with Laplace transforms. One example has me confused in the partial fractions step (bordered in red):
What is an $A$ doing there on the left-hand side of the numerator when performing the partial fractions step?
My understanding is that the equation should should go like this: $$Y(s)=\frac{1}{(s^2+1)(s^2+s)}+\frac{s+3}{s^2+s}$$ $$Y(s)=\frac{(s+3)(s^2+1)}{(s^2+1)(s^2+s)}$$ Factor out an $s$ in the denominator and do our partial fractions without some magic $A$ on the left: $$Y(s)=\frac{(s+3)(s^2+1)}{s(s+1)(s^2+1)}=\frac{A}{s}+\frac{B}{s+1}+\frac{Cs+D}{s^2+1}$$ And the rest of the steps appear to follow out much like the notes and the magic $A$ has no purpose unless I'm wrong? What was the point of it?
Please help or I will drink water and eat healthy.
It says '$+4$' not '$+A$'. This is because you incorrectly wrote $Y(s)$; $$Y(s)=\frac{1+(s+3)(s^2+1)}{(s^2+1)(s^2+s)}$$