Question

Determine the sum of the order and the degree of the differential equation given by: \[ y = x\frac{dy}{dx} + \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \]

• A. \( 2 \)
• B. \( 4 \)
• C. \( 3 \)
• D. \( 1 \)

Hint:

Never read the degree of a differential equation directly if any derivative sits inside a radical sign, fraction, or non-polynomial function. Always isolate and clear the radical term by squaring or multiplying out denominators first.

Answer 1

The Correct Option is C

Concept: The structural properties of a differential equation are defined as follows:

• Order: The highest derivative present in the equation.
• Degree: The power index of the highest-order derivative, provided the equation is expressed as a rational polynomial in terms of its derivatives. Fractional radicals wrapping derivative terms must be cleared before determining the degree.

Step 1: Isolate and clear the radical expression from the equation.
Move the polynomial terms to the left-hand side to isolate the square root: \[ y - x\frac{dy}{dx} = \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \] Square both sides of the equation to clear the radical fraction: \[ \left( y - x\frac{dy}{dx} \right)^2 = 1 + \left(\frac{dy}{dx}\right)^2 \]

Step 2: Expand the polynomial and determine the order and degree.
Expanding the left side using the binomial identity gives: \[ y^2 + x^2\left(\frac{dy}{dx}\right)^2 - 2xy\frac{dy}{dx} = 1 + \left(\frac{dy}{dx}\right)^2 \] The highest derivative appearing in this expanded expression is the first derivative \( \frac{dy}{dx} \), which gives an Order = 1. The highest power exponent attached to this derivative component is 2, which gives a Degree = 2.

Step 3: Calculate the final requested sum value.
Sum the isolated parameters: \[ \text{Sum} = \text{Order} + \text{Degree} = 1 + 2 = 3 \]