Text: Sections 1.3, 1.4
Direction Fields. A useful technique that can be used to visualize solutions of the first order DE
is the use of the direction field of the equation which consists of a short line segment of slope f(x,y) centered at the point (x,y). For example,
is the direction field of the differential equation
Using this direction field one can sketch solutions of the DE. For example, the solution with y(0)=.5 has the graph
A systematic way to construct the direction field is to draw a short line segment of slope m at various points of the curve f(x,y)=m for a selected range of values of m. These curves are the isoclines for the differential equation . For the above DE the isoclines are the curves .
The Phase Line. The first order DE is called autonomous since the independent variable does not appear explicitly. The isoclines are made up of horizonal lines y=f(m). The DE has the constant solution y=m if and only if f(m)=0. These values of m are the equilibrium or stationary points of the DE. The equilibrium point y=m is called a source if f(y) changes sign from - to + as y increases from just below y=m to just above y=m and is called a sink if f(y) changes sign from + to - as y increases from just below y=m to just above y=m; it is called a node if there is no change in sign.Solutions y of the DE appear to be attracted by the line the line y=m if m is a sink and move away if m is a source. The y-axis on which is plotted the equilibrium points of the DE with arrows between these points to indicate when the solution y is increasing or decreasing is called the phase line of the DE.
The autonomous DE has 0 and 1 as equilibrium points. The point y=0 is a source and y=2 is a sink. The direction field of this DE is
This DE is a logistic model for a population having 2 as the size of a stable population. If the population is reduced at a constant rate s>0, the DE becomes
which has a source at the larger of the two roots of the equation for s<2. If s>2 there is no equilibrium point and the popuation dies out as y is always decreasing. The point s=2 is called a bifurcation point of the DE.