Ordinary Differential Equations/Isoclines 1

First Order Differential Equations

Isoclines

An isocline uses another way to eliminate a variable. Instead of calculating y', we set it equal to a constant. Then we solve for y and graph the resulting equation. Frequently you'll see several different constants superimposed onto one graph, so you can see how different values compare.

Isocline is derived from the Greek words for same slope. An isocline is a line which joins neighbouring points with the same gradient, much like a contour line on a map joins all neighbouring points of the same height. What this means is that the isocline represents what happens to a point on the solution curve when the boundary conditions are changed, i.e how the solutions change when C is changed. When we consider many isoclines, it can help us to understand how the solutions "morph" on to each other as C changes.

Example 1: y'=f(x)

Fig. 1: Slope field (black) and some solutions (red) of y'=x

Fig. 2: Slope field (black), some solutions (red) and isoclines (blue) of y'=x

When the derivative of the dependent variable is set equal to a function of the independent variable, so

${\frac {dy}{dx}}=f(x),$

then the slope of the solution depends only on x. This means that the slope field of the DE will change only left to right - the slope will not vary from a point to the point directly above or below it. This can be seen in Fig. 1, showing the slope field of

${\frac {dy}{dx}}=x$

and some solutions. As you can see, the different solutions corresponding to different values of C lie directly above and below each other. The isoclines of this DE are given by setting y' equal to a constant:

${\frac {dy}{dx}}=k$

From the original DE, we now have:

$x=k\,$

Fig. 2 shows a set of isoclines corresponding to a range of k superimposed on Fig 1. We can see that these are vertical lines. This means that the locus of a point on a solution curve will be the corresponding isocline as C varies. Therefore, as C varies, a point on the solution curve will move vertically up and down, and so the entire curve will move up and down. There is, however, no guarantee that all points will move an equal amount for the same change in C.

The isoclines are vertical for all differential equations of the form

${\frac {dy}{dx}}=f(x).$

Example 2: y'=f(y)

Fig. 3: Slope field (black) and some solutions (red) of y'=y

Fig. 4: Slope field (black), some solutions (red) and isoclines (blue) of y'=y

A similar property is seen in DEs of the form

${\frac {dy}{dx}}=f(y),$

but the isoclines are horizontal. This is because the gradient of the solution is independent of the value of the x. For example, consider the solutions of

${\frac {dy}{dx}}=y,$

The solutions and the slope field of this DE are shown in Fig. 3. The isoclines are given by:

${\frac {dy}{dx}}=k$

From the original DE, we now have:

$y=k.\,$

A set of these are superimposed in Fig. 4. One can easily see that all solutions are just left-right translations of other solutions. By changing C, the solutions are moved side to side. However, the amount that they are moved is not linear for a linear change in C, and shown by the increasing spacings of the red curves. This information is not conveyed by the isoclines.

Horizontal isoclines are shared by all DEs of the form

${\frac {dy}{dx}}=f(y).$

Example 3: y'=f(x,y)

Fig. 5: Slope field (black) and some solutions (red) of y'=xy

Fig. 6: Slope field (black), some solutions (red) and isoclines (blue) of y'=xy

The last possible kind of isocline is that belonging to the DE of the form

${\frac {dy}{dx}}=f(x,y).$

There is no catch-all rule for these like there is for the previous two examples, but the isoclines are still easily computed.

Consider the DE

${\frac {dy}{dx}}=xy$

The isoclines are given by the equation

$xy=k,\,$

Isoclines of various values of k are plotted in Fig. 6. You can see from this that a point of the solution curve will tend vertically towards and horizontally away from the origin (or vice versa, for C changing the other way). This means that the solutions will become increasingly flat-bottomed/topped and wide or narrow and pointed as C changes.

The shape of the isoclines for this kind of DE depends on the original equation.