Find the order of the following differential equations:
`((d^(2)y)/(dx^(2)))^(2) + 3((dy)/(dx))^(3) = 5 x^(2)`

Answer

To find the order of the given differential equation:

\[ \left(\frac{d^2y}{dx^2}\right)^2 + 3\left(\frac{dy}{dx}\right)^3 = 5x^2 \]

we will follow these steps:

### Step 1: Identify the derivatives present in the equation.
In the given equation, we have two types of derivatives:
1. The second derivative \(\frac{d^2y}{dx^2}\)
2. The first derivative \(\frac{dy}{dx}\)

### Step 2: Determine the highest order of the derivatives.
The order of a derivative is determined by the highest number associated with the differentiation operator \(d\).

- The first derivative \(\frac{dy}{dx}\) has an order of 1.
- The second derivative \(\frac{d^2y}{dx^2}\) has an order of 2.

### Step 3: Find the highest order derivative.
Among the derivatives present:
- The highest order derivative is \(\frac{d^2y}{dx^2}\), which is of order 2.

### Conclusion: State the order of the differential equation.
Thus, the order of the given differential equation is:

\[ \text{Order} = 2 \]

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