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.
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
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 x2 + y2 = 4. (We need to specify above that y > 0. Why?)
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.
If (ln y) and y = e when x = 1, find the value of y greater than 1 that corresponds to x = e4.
SOLUTION: Separating, we get We integrate:
Using y = e when x = 1 yields so
When x = e4, we have thus ln2 y = 9 and ln y = 3 (where we chose ln y > 0 because y > 1), so y = e3.
Find the general solution of the differential equation
SOLUTION: We rewrite
Taking antiderivatives yields eu = ev + C, or u = ln(ev + c).