Page 44 - Schaum's Outline of Differential Equations

CHAP. 4] SEPARABLE FIRST-ORDER DIFFERENTIAL EQUATIONS 27

which can be algebraically simplified to

or

Equation (1) is separable; its solution is

The first integral is in In \y\. To evaluate the second integral, we use partial fractions on the integrand to
obtain

Therefore,

The solution to (_/) is in which can be rewritten as

where Substituting u = xly back into (2), we once again have (2) of Problem 4.12.

4.14. Solve

Phis differential equation is not separable. Instead it has the form y =f(x, y), with

where

so it is homogenous. Substituting Eqs. (4.6) and (4.7) into the differential equation as originally given, we obtain

which can be algebraically simplified to

or

Using partial fractions, we can expand (1) to

The solution to this separable equation is found by integrating both sides of (2). Doing so, we obtain In Ijcl -
2
In Ivl + In (v + 1) = c, which can be simplified to

2
2
Substituting v = ylx into (3), we find the solution of the given differential equation is x + y = ky.