Question

Classify the equation as separable, linear, exact, or none of these. Note that it is possible for the equation to have more than one classification

x⁵ydy+5dy=0

Answer 1

The given equation x⁵ydy + 5dy = 0 cannot be classified as separable, linear, or exact.

Step-by-step explanation:

The given equation is:

x⁵ydy + 5dy = 0

To classify the equation, we need to analyze its form:

• Separable equations are those that can be written in the form dy/dx = f(x)g(y).
• Linear equations are those that can be written in the form dy/dx + p(x)y = q(x).
• Exact equations are those that can be written in the form M(x, y)dx + N(x, y)dy = 0, where ∂M/∂y = ∂N/∂x.

Let's analyze the given equation:

1. We can rewrite the equation as x⁵ydy + 5dy/dx = 0.
2. This equation is not separable because it cannot be written in the form dy/dx = f(x)g(y).
3. This equation is not linear because it cannot be written in the form dy/dx + p(x)y = q(x).
4. This equation is not exact because it cannot be written in the form M(x, y)dx + N(x, y)dy = 0, where ∂M/∂y = ∂N/∂x.

Therefore, the given equation is none of these: separable, linear, or exact.

Answer 2

The equation x⁵ydy + 5dy = 0 is not linear or exact, but it is separable because the variables x and y can be moved to opposite sides of the equation.

Step-by-step explanation:

You are tasked with classifying the equation x⁵ydy + 5dy = 0 as separable, linear, exact, or none of these. This type of problem is common in differential equations, a topic covered in advanced high school or college mathematics courses. To classify the equation, you need to recognize the form of the differential equation and apply the definitions of separable, linear, and exact equations.

This given equation is not linear because a linear equation has the form y = mx + b or alternatively the differential form dy/dx + P(x)y = Q(x), where P(x) and Q(x) are functions of x only, and y does not appear in any products or powers.

The equation also does not fit the definition of an exact equation, which is of the form M(x, y)dx + N(x, y)dy = 0, where ∂M/∂y = ∂N/∂x. However, the given equation can be classified as separable because it can be rewritten in the form (ydx + 5dx) / x⁵y = -dy/y, where the variables x and y can be separated onto opposite sides of the equation.