Question

The nature of the differential equation \[ (x - y) \frac{dy}{dx} = x + 2y \] will be:

• A. Multipower
• B. Power one and linear
• C. Homogeneous and power zero
• D. Homogeneous and power one

Hint:

In a first-order linear differential equation, if the degree of both \(x\) and \(y\) is one and there is no constant term, the equation is homogeneous.

Answer 1

The Correct Option is D

The given differential equation is: \[ (x - y) \frac{dy}{dx} = x + 2y. \]

Step 1: Rearrange the equation.
We can rearrange the equation as: \[ \frac{dy}{dx} = \frac{x + 2y}{x - y}. \]

Step 2: Identify the type of equation.
This is a first-order linear differential equation with a non-constant coefficient. It is also a homogeneous equation because the right-hand side is a linear combination of \(x\) and \(y\), and both terms on the right-hand side have degree one in \(x\) and \(y\).

Step 3: Homogeneous and power one.
The equation is homogeneous because both the terms \(x\) and \(y\) appear to the first power, and there is no constant term that does not depend on \(x\) or \(y\). It is also of power one because the degree of both \(x\) and \(y\) is one.

Step 4: Conclusion.
Thus, the correct answer is (D) Homogeneous and power one.