Question

Find the degree of the differential equation \[xy \frac{d^2y}{dx^2} + x \left( \frac{dy}{dx} \right)^2 - y \left( \frac{dy}{dx} \right) = 2\]

Hint:

{Key Points:}

• Degree = Power of highest order derivative
• Equation must be polynomial in derivatives
• Here, highest order = \(\frac{d^2y}{dx^2}\) with power 1 → Degree = 1

Answer 1

Step 1: Recall the definition of degree of a differential equation The degree of a differential equation is defined as the power of the highest order derivative present in the equation, provided the equation is polynomial in all derivatives.
Step 2: Identify the highest order derivative The given differential equation is: \[xy \frac{d^2y}{dx^2} + x \left( \frac{dy}{dx} \right)^2 - y \left( \frac{dy}{dx} \right) = 2\] The highest order derivative present is \(\frac{d^2y}{dx^2}\) (second order derivative).
Step 3: Check if the equation is polynomial in derivatives The equation contains:

• \(\frac{d^2y}{dx^2}\) with power 1
• \(\left( \frac{dy}{dx} \right)^2\) with power 2
• \(\frac{dy}{dx}\) with power 1

All derivatives appear as polynomial terms. The equation is polynomial in all derivatives.
Step 4: Find the degree The highest order derivative is \(\frac{d^2y}{dx^2}\) and its power is 1. Therefore, the degree of the differential equation is: \[ \boxed{1} \] Note: The degree is 1 even though the equation contains \(\left( \frac{dy}{dx} \right)^2\) because the degree is determined by the power of the highest order derivative, not the lower order derivatives.