(a)
(b)
(a)
(b)
(B) goes with Figure N9–3a. The general solution is the family of parabolas y =
x^2 + C.
For (C) the slope field is shown in Figure N9–3b. The general solution is the
family of cubics y = x^3 − 3x + C.
(D) goes with Figure N9–3d; the general solution is the family of lines
.
Example 4 __
Verify that relations of the form x^2 + y^2 = r^2 are solutions of the d.e. .
Using the slope field in Figure N9–4 and your answer to (a), find the
particular solution to the d.e. given in (a) that contains point (4,−3).Figure N9–4SOLUTIONS:
By differentiating equation x^2 + y^2 = r^2 implicitly, we get , from
which , which is the given d.e.
x^2 + y^2 = r^2 describes circles centered at the origin. For initial point (4,−3),
(4)^2 + (−3)^2 = 25. So x^2 + y^2 = 25. However, this is not the particular
solution.
A particular solution must be differentiable on an interval containing the initial