Page 39 - Schaum's Outline of Differential Equations

22 SEPARABLE FIRST-ORDER DIFFERENTIAL EQUATIONS [CHAR 4

REDUCTION OF HOMOGENEOUS EQUATIONS
The homogeneous differential equation

having the property that/(fct, ty) =f(x, y) (see Chapter 3) can be transformed into a separable equation by making
the substitution

along with its corresponding derivative

The resulting equation in the variables v and x is solved as a separable differential equation; the required solution
to Eq. (4.5) is obtained by back substitution.
Alternatively, the solution to (4.5) can be obtained by rewriting the differential equation as

and then substituting

and the corresponding derivative

into Eq. (4.8). After simplifying, the resulting differential equation will be one with variables (this time, u and y)
separable.
Ordinarily, it is immaterial which method of solution is used (see Problems 4.12 and 4.13).
Occasionally, however, one of the substitutions (4.6) or (4.9) is definitely superior to the other one. In such
cases, the better substitution is usually apparent from the form of the differential equation itself. (See
Problem 4.17.)

Solved Problems

1
4.1. Solve x dx -y dy = 0.
2
For this differential equation, A(x) = x and B(y) = -y . Substituting these values into Eq. (4.2), we have

3
which, after the indicated integrations are performed, becomes x*/2 — ;y /3 = c. Solving for y explicitly, we obtain
the solution as