Question

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

Short answer

The given differential equation is of the first order and first degree, which corresponds to option (B). This was determined by rewriting the equation as a homogeneous differential equation and analyzing its order and degree.

Step-by-step solution

Differentiate with respect to x

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

Isolate the derivative term

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

Determine the order and degree of the differential equation

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.

Rewrite the given equation as a 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: \[ \frac{dy}{dx} = -\frac{A}{B} \frac{x}{y} \]

Determine the order and degree from the rewritten equation

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.

Key concepts

Order of Differential Equation

The order of a differential equation relates to the highest derivative present in the equation. In simple terms, it's like asking, "how many times do we differentiate a function to get the highest derivative?"

For example, if you have an equation involving \(\frac{d^2y}{dx^2}\), then the order is 2, because we differentiated twice. However, in our problem, after we simplified and rewrote it correctly, we found the order to be 1, based on the highest derivative \(\frac{dy}{dx}\).

The order is crucial because it tells us about the complexity and behavior of the differential equation. Understanding the order helps in selecting appropriate solution methods and understanding the nature of the equation.

Degree of Differential Equation

The degree of a differential equation is a bit different from the order. It is the power of the highest order derivative after the equation is free from fractions and radicals involving derivatives.

Let's say you have a differential equation like \(\left(\frac{d^2y}{dx^2}\right)^2 - 2x \frac{dy}{dx} = 0\). Here the order is 2, but the degree is determined by the exponent on \(\frac{d^2y}{dx^2}\), which is 2. However, in our example, the equation \(\frac{dy}{dx} = -\frac{A}{B} \frac{x}{y}\) has no powers on the derivative, so its degree is 1.

Understanding the degree of a differential equation is necessary for solving and analyzing its behavior, just like the order. Together, these concepts form the foundation for more advanced understandings of differential equations.

Implicit Differentiation

Implicit differentiation is a powerful technique when you have equations where variables are intricately connected and not easily separable. Instead of solving for one variable in terms of another, we differentiate the whole equation directly.

For example, in our exercise where \(Ax^2 + By^2 = C\), treating y as an implicit function of x allows us to differentiate directly: \(2Ax + 2B \left( \frac{dy}{dx} \right) y = 0\). This reveals the derivative \(\frac{dy}{dx}\) without needing to separate y from x.

This method is especially helpful in situations where separating variables is either complex or impossible. So, when faced with implicit forms, implicit differentiation becomes your go-to solution tool.