Question

The differential equation \[ y^2 \, dx + (3xy - 1) \, dy = 0 \] is:

• A. linear in \( y \)
• B. not a linear equation
• C. a homogeneous equation
• D. linear in \( x \)

Hint:

When analyzing differential equations, check for linearity and homogeneity by inspecting the terms and their degrees.

Answer 1

The Correct Option is B

We are given the differential equation: \[ y^2 \, dx + (3xy - 1) \, dy = 0 \] Step 1: Check if the equation is linear.
A linear differential equation must be of the form: \[ A(x) \frac{dy}{dx} + B(x) y = C(x) \] In this case, the equation involves terms like \( y^2 \) and \( 3xy \), which are nonlinear terms. Therefore, this is not a linear equation.
Step 2: Check if the equation is homogeneous.
A homogeneous differential equation is one in which all terms are of the same degree. This equation is not homogeneous either because it involves \( y^2 \) and \( 3xy \), which are not of the same degree.
Thus, the equation is not linear and not homogeneous.
Thus, the correct answer is: \[ \boxed{\text{not a linear equation}} \]