Step 1: Understanding the Question:
The objective is to find the degree of a given differential equation.
The degree of a differential equation is the highest power of the highest-order derivative appearing in the equation, provided the equation is in a polynomial form of its derivatives.
Before determining the degree, any radicals (square roots, cube roots, etc.) or fractional powers involving the derivatives must be eliminated by algebraic manipulation.
Step 2: Key Formula or Approach:
Identify the highest-order derivative present in the equation.
Rearrange the equation to isolate terms with radicals.
Eliminate the radical by squaring (or applying the necessary power) both sides.
Identify the power of the highest-order derivative in the resulting polynomial form.
Step 3: Detailed Explanation:
The given differential equation is: \(y = x\frac{dy}{dx} + a \sqrt{1 + \left(\frac{dy}{dx}\right)^2}\).
Here, the highest order derivative is \(\frac{dy}{dx}\), so the order of the differential equation is 1.
To find the degree, we must remove the square root. First, isolate the radical term:
\[ y - x\frac{dy}{dx} = a \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \]
Now, square both sides of the equation to eliminate the radical:
\[ \left(y - x\frac{dy}{dx}\right)^2 = \left(a \sqrt{1 + \left(\frac{dy}{dx}\right)^2}\right)^2 \]
Expanding the left side and simplifying the right side:
\[ y^2 + x^2\left(\frac{dy}{dx}\right)^2 - 2xy\frac{dy}{dx} = a^2 \left[ 1 + \left(\frac{dy}{dx}\right)^2 \right] \]
This equation is now a polynomial in \(\frac{dy}{dx}\). The highest derivative is \(\frac{dy}{dx}\), and its highest power (exponent) in this polynomial form is 2.
Therefore, the degree of the differential equation is 2.
Step 4: Final Answer:
The highest order derivative is of order 1, and after rationalizing the radical term, its highest power becomes 2. Thus, the degree is 2.