Barrons AP Calculus - David Bock

A first-order d.e. in x and y is separable if it can be written so that all the terms involving y are on
one side and all the terms involving x are on the other.
A differential equation has variables separable if it is of the form

The general solution is

EXAMPLE 8

Solve the d.e. given the initial condition y(0) = 2.

SOLUTION: We rewrite the equation as y dy = −x dx. We then integrate, getting

Since y(0) = 2, we get 4 + 0 = C; the particular solution is therefore x^2 + y^2 = 4. (We need to

specify above that y > 0. Why?)

EXAMPLE 9

If and t = 0 when s = 1, find s when t = 9.

SOLUTION: We separate variables:

then integration yields

Using s = 1 and t = 0, we get so C = + 2. Then

When t = 9, we find that s1/2 = 9 + 1, so s = 100.

EXAMPLE 10

If (ln y) and y = e when x = 1, find the value of y greater than 1 that corresponds to x = e^4.

SOLUTION: Separating, we get We integrate:

Using y = e when x = 1 yields so