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:
- \(\frac{d^2y}{dx^2}\), which is the second derivative of \(y\) with respect to \(x\).
- \(\frac{dy}{dx}\), which is the first derivative of \(y\) with respect to \(x\).

### Step 2: Determine the order of each derivative
- The order of \(\frac{d^2y}{dx^2}\) is 2 (since it is the second derivative).
- The order of \(\frac{dy}{dx}\) is 1 (since it is the first derivative).

### Step 3: Identify the highest order derivative
The order of a differential equation is defined as the order of the highest derivative present in the equation. In this case, the highest order derivative is \(\frac{d^2y}{dx^2}\), which is of order 2.

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

\[ \text{Order} = 2 \]

---