Here I will provide a typical example about how to Solve Linear First Order Differential Equations without y.
Example: Solve dy/dx=x^3+4x^2+x where y(0)=4
Solution (Step by Step):
Step 1: Move dx to the right side: dy=x^3 dx - 4x^2 dx + xdx
Step 2: Do the integration and you will get this: y=(x^4)/4 - (4x^3)/3 + (x^2)/2 + C
Step 3: Since y(0)=4 is given, we pluging x=0 and y=3 and we get this:
Therefore the full solution is
y=(x^4)/4 - (4x^3)/3 + (x^2)/2 + 4
Basically all you do is move the dx to the other side, do integration and then solve for C using the initial condition given.