Question

Classify each differential equation as separable, exact, linear, homogeneous, or Bernoulli. Some equations may belong to more than one category. Do not solve.

Answer 1

Differential equations can be classified as separable, exact, linear, homogeneous, or Bernoulli. Each type has specific characteristics and can be identified based on the form of the equation. For example, separable equations can be separated into variables that can be integrated separately, while exact equations can be expressed as the derivative of a function.

Step-by-step explanation:

A differential equation can be classified as separable, exact, linear, homogeneous, or Bernoulli. Let's define each type:

1. Separable: In a separable differential equation, the variables can be separated and each variable can be integrated separately. An example of a separable equation is dy/dx = (x + y)/x.
2. Exact: An exact differential equation can be expressed as the derivative of a function. The equation M(x, y)dx + N(x, y)dy = 0 is exact if (∂M/∂y) = (∂N/∂x).
3. Linear: A linear differential equation is of the form dy/dx + p(x)y = g(x). It is linear because y and its derivative are only raised to the first power.
4. Homogeneous: A homogeneous differential equation is of the form dy/dx = f(y/x) or y = x(m+1)f(y/x). It is homogeneous because all terms have the same degree.
5. Bernoulli: A Bernoulli differential equation is of the form dy/dx + p(x)y = q(x)y^n where n ≠ 0 and n ≠ 1.