Step 1: Understanding the Concept:
A differential equation is characterized by its "Order" and its "Degree."
The Order of a differential equation is the order of the highest derivative occurring in the equation. It is a straightforward property that can be identified by looking for the highest power of \(d\) in the derivative terms (e.g., \(d^2y/dx^2\) is order 2).
The Degree of a differential equation is the power of the highest order derivative when the equation is expressed as a polynomial in derivatives. This means the equation must be free from radicals (roots) or fractional powers affecting the derivative terms. If the equation contains derivatives inside functions like \(\sin, \cos, \log,\) or under square roots, it must be rationalized before determining the degree. If it cannot be expressed as a polynomial, the degree is said to be "not defined."
Key Formula or Approach:
1. Inspect the equation for the highest order derivative to find the Order.
2. Manipulate the equation algebraically to remove any radicals involving derivatives.
3. Once in polynomial form, find the exponent of the highest order derivative to find the Degree.
4. Sum the results as requested.
Step 2: Detailed Explanation:
The given equation is \(y = x \frac{dy}{dx} + \sqrt{1 + \left(\frac{dy}{dx}\right)^{2}}\).
By inspection, the only derivative present is the first derivative \(\frac{dy}{dx}\). Therefore, the highest order derivative is of order 1.
Order = 1.
To find the degree, we observe a radical term \(\sqrt{1 + (dy/dx)^2}\). We must clear this square root.
First, isolate the radical by moving the other terms to the left-hand side:
\[ y - x \frac{dy}{dx} = \sqrt{1 + \left(\frac{dy}{dx}\right)^{2}} \]
Now, square both sides of the equation to eliminate the root:
\[ \left(y - x \frac{dy}{dx}\right)^{2} = 1 + \left(\frac{dy}{dx}\right)^{2} \]
Expanding the left side using the identity \((a-b)^2 = a^2 - 2ab + b^2\):
\[ y^{2} - 2xy \frac{dy}{dx} + x^{2} \left(\frac{dy}{dx}\right)^{2} = 1 + \left(\frac{dy}{dx}\right)^{2} \]
Rearranging terms to form a standard polynomial in terms of \(\frac{dy}{dx}\):
\[ (x^{2} - 1) \left(\frac{dy}{dx}\right)^{2} - 2xy \left(\frac{dy}{dx}\right) + (y^{2} - 1) = 0 \]
In this polynomial form, the highest order derivative is \(\frac{dy}{dx}\). The highest power (exponent) to which this derivative is raised is \(2\).
Degree = 2.
The question asks for the sum of the Order and the Degree:
\[ \text{Sum} = \text{Order} + \text{Degree} = 1 + 2 = 3 \]
Step 3: Final Answer:
The order of the equation is \(1\) and, after clearing the radical, the degree is found to be \(2\). The sum is \(3\).