A differentiable function--and the solutions to differential equations better be differentiable--has tangent lines at every point. Let's draw small pieces of some of these tangent lines of the function :
The slope, y'(x), of the solutions y(x), is determined once we know the values for x and y , e.g., if x=1 and y=-1, then the slope of the solution y(x) passing through the point (1,-1) will be . If we graph y(x) in the x-y plane, it will have slope 2, given x=1 and y=-1. We indicate this graphically by inserting a small line segment at the point (1,-1) of slope 2.
Of course, doing this at just one point does not give much information about the solutions. We want to do this simultaneously at many points in the x-y plane.
Let's consider the following differential equation:
Here, the right-hand side of the differential equation depends only on the dependent variable y, not on the independent variable x. Such a differential equation is called autonomous. Autonomous differential equations are always separable.