Advanced High-School Mathematics

SECTION 5.5 Differential Equations 309

which describes a rapidly-
decreasing exponential func-
tion ofx. The slope field, to-
gether with the particular so-
lution with the initial condi-
tiony(0) = 2 is indicated to
the right.

Some first-order ODE are not separable as they stand, but through
a change of variables can be transformed into a separable ODE. Such
is the case of ODE of the form

dy

dx

= F

Çy

x

å

, (5.2)

for some functionF. A change of independent variable

v =

y
x
will accomplish this. To see this, we note that

y = vx,

dy

dx

= x

dv

dx

+v;

with respect toxandvthe ODE (5.2) becomes

x

dv

dx

+v = F(v).

The variablesxandvseparate easily, resulting in the ODE

1

F(v)−v

dv

dx

=

1

x

,

which can be solved in principle as above.

Example 2. The first-order ODE