Question

Degree of the differential equation $y = x\frac{dy}{dx} + a\sqrt{1 + (\frac{dy}{dx})^2}$ is

• A. 4
• B. 3
• C. 1
• D. 2

Hint:

Radical sign over a derivative? Square it! The degree is only defined for differential equations in polynomial form.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The degree of a differential equation is the power of the highest order derivative, provided the equation is a polynomial in its derivatives. To find the degree, we must first eliminate radicals (square roots) and fractions involving derivatives.
Step 2: Key Formula or Approach:
1. Isolate the term with the square root.
2. Square both sides to rationalize the equation.
Step 3: Detailed Explanation:
Isolate the radical: \[ y - x \frac{dy}{dx} = a \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \] Square both sides: \[ \left( y - x \frac{dy}{dx} \right)^2 = a^2 \left[ 1 + \left( \frac{dy}{dx} \right)^2 \right] \] Expanding the left side: \[ y^2 + x^2 \left( \frac{dy}{dx} \right)^2 - 2xy \frac{dy}{dx} = a^2 + a^2 \left( \frac{dy}{dx} \right)^2 \] Now the equation is a polynomial in \(\frac{dy}{dx}\). The highest order derivative is \(\frac{dy}{dx}\) (Order 1), and its highest power in this polynomial form is 2.
Step 4: Final Answer:
The degree of the differential equation is 2.