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:

The degree is determined by the power of the highest order derivative only. Do not get confused by higher powers on lower order derivatives, such as the power of 2 on \( \frac{dy}{dx} \) in this problem. The degree is 1, not 2.

Answer 1

Step 1: Understanding the Concept:
The order of a differential equation is the order of the highest derivative present in the equation. The degree of a differential equation is the highest power (positive integer exponent) of the highest order derivative, after the equation has been cleared of any radicals or fractional powers of the derivatives.
Step 2: Key Approach:
1. Identify the highest order derivative in the equation.
2. Check if the equation is a polynomial in its derivatives. If not, manipulate it to become one.
3. Find the power of the highest order derivative. This power is the degree.
Step 3: Detailed Explanation:
The given differential equation is: \[ xy \frac{d^2y}{dx^2} + x \left(\frac{dy}{dx}\right)^2 - y\left(\frac{dy}{dx}\right) = 2 \] 1. Identify the derivatives and their orders:
\( \frac{dy}{dx} \) is the first-order derivative.
\( \frac{d^2y}{dx^2} \) is the second-order derivative.
The highest order derivative is \( \frac{d^2y}{dx^2} \), so the order of the equation is 2.
2. Check for polynomial form:
The equation is already expressed as a polynomial in terms of its derivatives \( \frac{d^2y}{dx^2} \) and \( \frac{dy}{dx} \). There are no radicals or fractional exponents on the derivative terms.
3. Determine the degree:
We need to find the highest power of the highest order derivative, which is \( \frac{d^2y}{dx^2} \). The term containing the highest order derivative is \( xy \frac{d^2y}{dx^2} \). The power of \( \frac{d^2y}{dx^2} \) in this term is 1. \[ xy \left(\frac{d^2y}{dx^2}\right)^1 + x \left(\frac{dy}{dx}\right)^2 - y\left(\frac{dy}{dx}\right)^1 = 2 \] The highest power of \( \frac{d^2y}{dx^2} \) is 1.
Step 4: Final Answer:
The degree of the differential equation is 1.