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

Answer

To find the degree of the given differential equation:

**Given Differential Equation:**
\[ x^2 \left( \frac{d^2y}{dx^2} \right)^3 + y \left( \frac{dy}{dx} \right)^4 + x^3 = 0 \]

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

**Step 2: Determine the order of the highest derivative.**
The order of the highest derivative \(\frac{d^2y}{dx^2}\) is 2, as it is the second derivative of \(y\).

**Step 3: Identify the power of the highest derivative.**
In the equation, the highest derivative \(\frac{d^2y}{dx^2}\) is raised to the power of 3:
\[ \left( \frac{d^2y}{dx^2} \right)^3 \]

**Step 4: Define the degree of the differential equation.**
The degree of a differential equation is defined as the power of the highest derivative when the equation is a polynomial in derivatives.

**Conclusion:**
Since the highest 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 differential equation is **3**.

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