Question

The ordinary differential equation \[ \frac{dy}{dx} = x^2 y \] has \( y \) as the dependent variable and \( x \) as the independent variable. Which of the following classification(s) is/are applicable to the equation?

• A. Linear
• B. Non-linear
• C. First order
• D. Second order

Hint:

A first-order differential equation involves only the first derivative. If the equation includes products of the dependent variable and its derivatives, it is non-linear.

Answer 1

The Correct Option is A, C

The given equation is a first-order differential equation because it involves only the first derivative of \( y \) with respect to \( x \). Step 1: Classification of Order
The order of a differential equation is determined by the highest derivative present. Since the highest derivative here is \( \frac{dy}{dx} \), the equation is a first-order equation. Step 2: Linear or Non-linear
This equation is a non-linear differential equation because the dependent variable \( y \) is multiplied by its derivative, making the equation non-linear. If the equation involved only linear terms (e.g., \( y' + p(x)y = q(x) \)), it would have been linear. Final Answer: \[ \boxed{\text{(C) First order, (B) Non-linear}} \]