Question

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

Hint:

The degree of a differential equation is determined by the highest power of the highest order derivative in the equation, provided the equation is polynomial in derivatives.

Answer 1

The degree of a differential equation is defined as the power of the highest order derivative, provided the equation is polynomial in derivatives. First, let's rewrite the given equation: \[ x y \left( \frac{d^2 y}{dx^2} \right)^2 + x \frac{dy}{dx} - y = 2. \]

Step 1: Identify the highest order derivative. In this equation, the highest order derivative is \( \frac{d^2 y}{dx^2} \), which is the second-order derivative.

Step 2: Make sure the equation is polynomial in the derivatives. We need to ensure that the equation does not contain fractional or irrational powers of derivatives. In the given equation, \( \left( \frac{d^2 y}{dx^2} \right)^2 \) is a polynomial expression in \( \frac{d^2 y}{dx^2} \), and there are no fractional or irrational powers of the derivative.

Step 3: Identify the degree of the highest order derivative. The term \( \left( \frac{d^2 y}{dx^2} \right)^2 \) involves squaring the second derivative, so the degree of the highest order derivative is 2. Conclusion: The degree of the given differential equation is \( \boxed{2} \).