Page 41 - Schaum's Outline of Differential Equations

24 SEPARABLE FIRST-ORDER DIFFERENTIAL EQUATIONS [CHAR 4

or, by evaluating,

Since it is impossible algebraically to solve this equation explicitly for y, the solution must be left in its present
implicit form.

2
4.6. Solve dy = 2t(y + 9) dt.
This equation may be rewritten as

which is separable in variables y and t. Its solution is

or, upon evaluating the given integrals,

Solving for y, we obtain

or
with k = 3c.

4.7. Solve

This equation may be rewritten in differential form

which is separable in the variables x and t. Its solution is

Evaluating the first integral by first completing the square, we obtain

or arctan (x— l) — t=c
Solving for x as a function of t, we obtain
arctan (x—l) = t+c
x - 1 = tan (t + c)
or x=l +tan(t + c)