The differential equation whose solution is \(\mathrm{Ax}^{2}+\mathrm{By}^{2}=1\), where \(\mathrm{A}\) and \(\mathrm{B}\) are arbitrary constants is of. (A) second order and second degree (B) first order and first degree (C) first order and second degree (D) second order and first degree

We'll take the partial derivative of the given equation with respect to x. \[ A(2x) + B(0) = 0 \] This simplifies to: \[ 2Ax = 0 \]

We want to isolate the derivative term, in this case \(x\): \[ x = 0 \] Now, let's analyze the result:

Looking at the final equation, we can determine: - Order: The order of a differential equation is the highest derivative appearing in the equation. In this case, there is no derivative term present in the equation, hence its order is 0. - Degree: The degree of a differential equation is the highest power of the highest order derivative present. Since there is no derivative term here, the degree is not applicable. However, if you recall the options, there is no option with an order of 0, meaning we may have made a mistake during our calculation. In step 1, we took partial derivatives considering A and B as constants, but they are arbitrary constants. So step 1 was not the most appropriate way to approach this problem. Let's try another approach by considering the given equation as a homogeneous differential equation.

Let's consider the given equation as an implicit form of the following homogeneous differential equation: \[ Ax^2 + By^2 = C \] We can rewrite this equation as: \[ Ax^2 + By^2 - C = 0 \] Now, we will differentiate this equation with respect to x, treating y as an implicit function of x (i.e., \(y = y(x)\)). The result will be: \[ 2Ax + 2B [\frac{dy}{dx}] y = 0 \] This can be written as: \[ 2Ax = -2B [\frac{dy}{dx}] y \] Then we can simplify this differential equation to: \[ \frac{dy}{dx} = -\frac{A}{B} \frac{x}{y} \]

Looking at our rewritten differential equation, we can determine the following: - Order: The highest derivative term appearing in the equation is \(\frac{dy}{dx}\), so the order is 1. - Degree: The highest power of the highest order derivative term is 1 (since there is no exponent applied to \(\frac{dy}{dx}\)), therefore the degree is 1. So, the given differential equation is of the first order and first degree. This corresponds to option (B) in the given choices.