The isocline method of constructing a phase portrait

We will give an approximate method of drawing phase portraits, that is finding the forms of solutions [x(t), y(t)] on the phase plane for the system (5.2). For the purpose, the system (5.2) will be written in a different form 

The following geometric interpretation of the solution of equation (A35) may be provided if we draw at each point (x, y) of the phase plane a section of a straight line of the slope equal to /(x,y), and if the trajectory of the solution [x(t), y(t)] crosses this point, then the trajectory is tangent to the section plotted at this point. 

The above remark forms the basis for an approximate method of constructing the phase portraits for equation (A35), called the isocline method. At the points (x, y) where the slope of the trajectory is a, the equality tan(a) = f(x, y) is fulfilled, since /[x, y(x)] = y (x). All such points lie on a line called the isocline, defined by the formula tan (a) = /(x,y) isoclines connect the points at which the sections tangent to the trajectory are identically inclined with respect to the x-axis. 

Let us draw, for example, an approximate phase portrait for the system of equations 

The lines dividing the phase plane into regions of a different course of trajectories are called separatrices. For example, the isocline of horizontal tangents, /(x,y) = 0, divides the phase plane into regions in which the solutions y(x) either increase or decrease it thus is a separatrix. We shall 

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