What is the degree of `x((d^(2)y)/(dx^(2)))^(3)+y((dy)/(dx))^(4)+x^(3)=0`?

Answer

To find the degree of the given differential equation \( x\left(\frac{d^2y}{dx^2}\right)^3 + y\left(\frac{dy}{dx}\right)^4 + x^3 = 0 \), we will follow these steps:

### Step 1: Identify the highest order derivative
The given equation contains two derivatives:
- The second derivative \( \frac{d^2y}{dx^2} \)
- The first derivative \( \frac{dy}{dx} \)

The highest order derivative present in the equation is \( \frac{d^2y}{dx^2} \), which is of order 2.

### Step 2: Look for the power of the highest order derivative
Next, we need to determine the power of the highest order derivative in the equation. In this case, the highest order derivative \( \frac{d^2y}{dx^2} \) is raised to the power of 3.

### Step 3: Check for rational powers
To find the degree of a differential equation, we must ensure that the power of the highest order derivative is a non-negative integer. In our case, the power of \( \frac{d^2y}{dx^2} \) is 3, which is indeed a non-negative integer.

### Step 4: Conclusion
Since the highest order derivative \( \frac{d^2y}{dx^2} \) is raised to the power of 3, the degree of the differential equation is 3.

Thus, the degree of the given differential equation is **3**.

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