14-5 THE LINEAR EQUATION OF FIRST ORDER 1111
curve of the second family through the point. By analytic geometry, perpendicular
lines have slopes m1 and m = - 1, n. Hence, a curve y = f(x) of the second
family satisfies the equation y' = - 1/F(x, y) at each point on its graph and,
therefore, the orthogonal trajectories form the general solution of the differential
equation y' = - 1/F(x, y) in D (at least, where F 0). One can also write the
given equation in the form P(x, y) dx + Q(x, y) dy = 0, and then the equation
of the orthogonal trajectories is -Q(x, y) dx + P(x, y) dy = 0; thus u = Pi + Qj
is replaced by = - i + Pj. (What is the geometric meaning of u?) We
observe that the first family is also the family of orthogonal trajectories of the
second family. Inportant examples of orthogonal trajectories are the sets of lines
parallel to the coordinate axes, and the curves r = const and 0 = const in polar
coordinates (in fact, orthogonal trajectories are often the basis for curvilinear
ctoordinates in the plane).
For each of the differential equations below, state the differential equation
for the orthogonal trajectories and find its general solution. Also graph the
solutions of the givien equation and the orthogonal trajectories.
(a)' = e (b) y' = x2 + 1 (c) y dx + xdy = (d) y dxI- xdy = 0
(e) x2 dy/ + y dx = 0 (1f) x + y) dx + (x - iy) dy = 0
t8. For an elquation y' = F(x)G(y) in which G has zeros at Yi and Y2, the lines y = Y,
and y = Y2 are solutions, and it appears that solutions starting between these
lines are "trapped" between them, as in Figure 14-10. To establish this, assume
that F(x) is continuous for all x, that G'(y) is continuous for all y and that
G(y) = Gi(y2) = 0, G(y) >0 for Yli K < Y2 Let Yi K K? 2 and show that
a solution y = f(x) with initial value (xo, Yo) is defined for - Oc < x ( c and
satisfies YI, f(x) 92 for all x. {Iint. Show that f is defined implicitly by the
equation g(y) = q (x), where
g(y) J= f dI, = JjF(t) dt
Show that there are positive constants K1, K2 such that G(y) K1(y - YI) and
G(y) < K2(y2 - y) for YI 9Y2 and, hence, conclude that g(y) is monotone
strictly increasing for - c c y < 0 with limit + C00 as y -- Y2-, limit - c0
as y > Y +. Now use the fact that f(x) = g-l(x).
49. Let P(x, y) and Q(x, y) have continuous first derivatives for all (x, y) and let P
and Q be homogeneous of degree n, so that P(tx, ty) = t"P(x, y), Q(tx, ty) =
tnQ(x, y). Show that the equation P dx + 9 dy = 0 can be written in the form
y' = g(y x) and that (xP + y9) - is an integrating factor for the equation.
110. Prove: if y = f(x) is a solution of the differential equation y' = g(y/x), then so
also is y = h(x) = (1/k)f(kx), where k is a nonzero constant. Interpret the result
as showing that the solutions of y' = g(y/x) are a family of similar curves.
14-5 THE LINEAR EQUATION OF FIRST ORDER
A first-order equation of form
s(x)y' + p(x)y = q(x)
