Understanding Order and Degree of Differential Equations

The question asks us to determine the order and degree of the given differential equation: \(y^2 = 4a(x - a)\), where 'a' is an arbitrary constant.

To find the order and degree, we must first form a differential equation by eliminating the arbitrary constant 'a'. The order of a differential equation is the order of the highest derivative present in the equation. The degree is the power of the highest derivative when the equation is a polynomial in derivatives.

Eliminating the Arbitrary Constant 'a'

We start with the given equation:

\(y^2 = 4a(x - a)\) (Equation 1)

To eliminate the single arbitrary constant 'a', we need to differentiate the equation once with respect to x.

Differentiating Equation 1 with respect to x:

\(\frac{d}{dx}(y^2) = \frac{d}{dx}(4a(x - a))\)

\(2y \frac{dy}{dx} = 4a(1 - 0)\)

\(2y \frac{dy}{dx} = 4a\)

Dividing by 2:

\(y \frac{dy}{dx} = 2a\)   (Equation 2)

Now we have two equations, Equation 1 and Equation 2, involving 'a'. We can eliminate 'a' using these equations.

From Equation 2, we can express 'a' as:

\(a = \frac{1}{2} y \frac{dy}{dx}\)

Substitute this expression for 'a' back into Equation 1:

\(y^2 = 4 \left(\frac{1}{2} y \frac{dy}{dx}\right) \left(x - \frac{1}{2} y \frac{dy}{dx}\right)\)

\(y^2 = 2y \frac{dy}{dx} \left(x - \frac{1}{2} y \frac{dy}{dx}\right)\)

Assuming \(y \neq 0\), we can divide both sides by y:

\(y = 2 \frac{dy}{dx} \left(x - \frac{1}{2} y \frac{dy}{dx}\right)\)

Expand the right side:

\(y = 2x \frac{dy}{dx} - 2 \frac{dy}{dx} \cdot \frac{1}{2} y \frac{dy}{dx}\)

\(y = 2x \frac{dy}{dx} - y \left(\frac{dy}{dx}\right)^2\)

Rearrange the terms to get the standard form of the differential equation:

\(y \left(\frac{dy}{dx}\right)^2 - 2x \frac{dy}{dx} + y = 0\)

Determining Order and Degree

Now that we have the differential equation, we can find its order and degree.

• Order: The order of a differential equation is determined by the highest order derivative present. In this equation, the only derivative is \(\frac{dy}{dx}\), which is the first derivative. Therefore, the order is 1.
• Degree: The degree of a differential equation is the power of the highest order derivative, provided the equation is a polynomial in the derivatives. The highest order derivative is \(\frac{dy}{dx}\). The equation \(y \left(\frac{dy}{dx}\right)^2 - 2x \frac{dy}{dx} + y = 0\) is a polynomial in \(\frac{dy}{dx}\). The highest power of \(\frac{dy}{dx}\) in this equation is 2 (from the term \(y \left(\frac{dy}{dx}\right)^2\)). Therefore, the degree is 2.

Thus, the order of the differential equation is 1 and the degree is 2.

Let's look at the options provided:

1. 1, 2
2. 2, 1
3. 2, 2
4. 1, 1

Our calculated order is 1 and degree is 2, which corresponds to the first option.

Summary of Finding Order and Degree

For a differential equation, the order is the level of differentiation (1st, 2nd, etc.) of the highest derivative. The degree is the exponent on that highest derivative term, after simplifying the equation to remove fractional or radical powers of derivatives.

Revision Table: Differential Equation Concepts

Additional Information on Differential Equations

Differential equations are mathematical equations that relate a function with its derivatives. They are used to model various phenomena in science and engineering.

• Ordinary Differential Equation (ODE): An equation involving derivatives of a function of a single independent variable. The problem discussed is an ODE.
• Partial Differential Equation (PDE): An equation involving partial derivatives of a function of multiple independent variables.
• Forming a Differential Equation: One common method is to eliminate arbitrary constants from a given relation involving variables and constants. The number of arbitrary constants usually determines the order of the resulting differential equation. In this case, there was one arbitrary constant 'a', leading to a first-order differential equation.
• Polynomial in Derivatives: An equation is a polynomial in derivatives if it can be written such that each term is a constant times a product of derivatives raised to non-negative integer powers. \(y (\frac{dy}{dx})^2 - 2x \frac{dy}{dx} + y = 0\) is a polynomial in \(\frac{dy}{dx}\) because the terms involving the derivative are \(y (\frac{dy}{dx})^2\) and \(-2x \frac{dy}{dx}\).

Understanding the order and degree is fundamental for classifying and solving differential equations.