The degree of the differential equation :
`xy((d^(2)y)/(dx^(2)))^(2)+x^(4)((dy)/(dx))^(3)-y(dy)/(dx)=0` is :

Options

A. 4
B. 3
C. 2
D. 1

Answer

To find the degree of the given differential equation:

\[ xy\left(\frac{d^2y}{dx^2}\right)^2 + x^4\left(\frac{dy}{dx}\right)^3 - y\frac{dy}{dx} = 0 \]

we will follow these steps:

### Step 1: Identify the highest derivative
The highest derivative in the given equation is \(\frac{d^2y}{dx^2}\).

### Step 2: Determine the power of the highest derivative
In the equation, the term \(\left(\frac{d^2y}{dx^2}\right)^2\) indicates that the highest derivative \(\frac{d^2y}{dx^2}\) is raised to the power of 2.

### Step 3: Define the degree of the differential equation
The degree of a differential equation is defined as the power of the highest derivative after the equation has been made polynomial in derivatives.

### Step 4: Check if the equation is polynomial in derivatives
The given equation is already polynomial in terms of the derivatives, as all the terms involving derivatives are expressed as powers.

### Step 5: Conclude the degree
Since the highest derivative \(\frac{d^2y}{dx^2}\) has a power of 2, the degree of the differential equation is:

\[ \text{Degree} = 2 \]

### Final Answer
The degree of the differential equation is **2**.

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