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

Given Differential Equation:
The given differential equation is
\[ xy \frac{d^2y}{dx^2} + x \left( \frac{dy}{dx} \right)^2 - y \left( \frac{dy}{dx} \right) = 2 \] Step 1: Identify the highest order derivative
In the given equation, the highest order derivative is \( \frac{d^2y}{dx^2} \), which is the second derivative of \(y\) with respect to \(x\). Thus, the order of the differential equation is 2.

Step 2: Determine the degree of the differential equation
The degree of a differential equation is the power of the highest order derivative after it has been made free from radicals and fractions.
In the given equation, the highest order derivative is \( \frac{d^2y}{dx^2} \). It appears without any fractions or radicals, and the equation is already polynomial in form with respect to the derivatives.

Therefore, the degree of the given differential equation is 1.

Conclusion:
The degree of the given differential equation is 1.