Can we solve $\frac{dx}{dt} = 3x+3t$ by "multiplying by $dt$"? In preparation for ODE, I was just doing some practice problems when I came across this.
$$\frac{dx}{dt} = 3x+3t$$
Although I looked at the solution where it states to do an integration factor, is there anything inherently wrong with simply multiplying dt on both sides and then taking the integral of both sides and then isolating $x$? I would appreciate any clarity on this matter. Thanks so much.

This is a linear ordinary differential equation to written as
$$\frac{dx}{dt}-3x=3t$$
its integrating factor is $e^{\int -3 dt}=e^{-3t}$, then
$$x=e^{3t}\int e^{-3t} 3t dt+C e^{3t}$$
The other way is to assume $x+t=v \implies \frac{dx}{dt}=\frac{dv}{dt}-1$,
then you get
$$\frac{dv}{dt}=3v+1 \implies \int \frac{dv}{3v+1}=\int dt +C$$
