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:

y(0)=4=0-0+0+C

C=4

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.