Page 32 - Schaum's Outline of Differential Equations

CHAP. 3] CLASSIFICATIONS OF FIRST-ORDER DIFFERENTIAL EQUATIONS 15

HOMOGENEOUS EQUATIONS
A differential equation in standard form (3.1) is homogeneous if

for every real number t. Homogeneous equations are solved in Chapter 4.
Note: In the general framework of differential equations, the word "homogeneous" has an entirely different
meaning (see Chapter 8). Only in the context of first-order differential equations does "homogeneous" have the
meaning defined above.

SEPARABLE EQUATIONS
Consider a differential equation in differential form (3.2). If M(x, y) = A(x) (a function only of x) and
N(x, y) = B(y) (a function only of y), the differential equation is separable, or has its variables separated.
Separable equations are solved in Chapter 4.

EXACT EQUATIONS

A differential equation in differential form (3.2) is exact if

Exact equations are solved in Chapter 5 (where a more precise definition of exactness is given).

Solved Problems

2
3.1. Write the differential equation xy' — y = 0 in standard form.
2
2
Solving for y', we obtain y' = y lx which has form (3.1) whh/(x, y) = y lx.
x
3.2. Write the differential equation e y' + e^y = sin x in standard form.
Solving for y', we obtain

or
which has form (3.1) wi\hf

5
3.3. Write the differential equation (/ + y) = sin (y'lx) in standard form.
This equation cannot be solved algebraically for y', and cannot be written in standard form.

3.4. Write the differential equation y(yy' - 1) = x in differential form.
Solving for y', we have