(a)
(b)
(c)
Derivatives of Implicitly Defined Functions
In Examples 2 and 3 above, each d.e. was of the form or y′ = f(x). We
were able to find the general solution in each case very easily by finding the
antiderivative .
We now consider d.e.’s of the form , where f(x,y) is an expression in x
and y; that is, is an implicitly defined function. Example 4 illustrates such a
differential equation. Here is another example.
Example 5 __
Figure N9–5 shows the slope field for
y′ = x + y. (1)At each point (x,y) the slope is the sum of its coordinates. Three particular
solutions have been added, through the points
(0,0)
(0, −1)
(0, −2)