Question

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

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

Hint:

To find degree, first remove radicals involving derivatives, then check the highest power of the highest order derivative.

Answer 1

The Correct Option is C

Concept: The degree of a differential equation is the highest power of the highest order derivative after removing radicals and fractions involving derivatives.

Step 1: Given: \[ y=x\frac{dy}{dx}+a\sqrt{1+\left(\frac{dy}{dx}\right)^2} \] Let: \[ p=\frac{dy}{dx} \] Then equation becomes: \[ y=xp+a\sqrt{1+p^2} \]

Step 2: Move \(xp\) to the left side. \[ y-xp=a\sqrt{1+p^2} \]

Step 3: Square both sides to remove the radical. \[ (y-xp)^2=a^2(1+p^2) \]

Step 4: Now the derivative \(p=\frac{dy}{dx}\) appears with highest power \(2\).
Therefore, the degree of the differential equation is: \[ 2 \] Hence, \[ \boxed{2} \]