Determine the system function H(s) of a system and find out the differential equation
Your transfer function looks correct. Note that you can rewrite $H(s)$ as

$$H(s)=-\frac{3s}{5s+3}\tag{1}$$

or, equivalently,

$$5sY(s)+3Y(s)=-3sX(s)\tag{2}$$

Since multiplication by $s$ corresponds to differentiation in the time domain you obtain from $(2)$

$$5y'(t)+3y(t)=-3x'(t)\tag{3}$$