Calculus AB and Calculus BC

CHAPTER 9 Differential Equations

D. SOLVING FIRST-ORDER DIFFERENTIAL EQUATIONS ANALYTICALLY

In the preceding sections we solved differential equations graphically, using slope fields, and numerically, using Euler’s method. Both methods yield approximations. In this section we review how to solve some differential equations exactly.

Separating Variables

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

Image

The general solution is

Image

EXAMPLE 8

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

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

Image

Since y(0) = 2, we get 4 + 0 = C; the particular solution is therefore x2 + y2 = 4. (We need to specify above that y > 0. Why?)

EXAMPLE 9

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

SOLUTION: We separate variables:

Image

then integration yields

Image

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

Image

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

EXAMPLE 10

If (ln y) Image and y = e when x = 1, find the value of y greater than 1 that corresponds to x = e4.

SOLUTION: Separating, we get Image We integrate:

Image

Using y = e when x = 1 yields Image so

Image

When x = e4, we have Image thus ln2 y = 9 and ln y = 3 (where we chose ln y > 0 because y > 1), so y = e3.

EXAMPLE 11

Find the general solution of the differential equation Image

SOLUTION: We rewrite Image

Taking antiderivatives yields eu = ev + C, or u = ln(ev + c).