Indicate the order of each of the following differential equations :
`log ((d^(2)y)/(dx^(2)))=((dy)/(dx))^(3)+x`.

Answer

To determine the order of the given differential equation:

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

we follow these steps:

### Step 1: Identify the derivatives present in the equation
In the equation, we have two derivatives:
- \(\frac{d^2y}{dx^2}\), 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 highest order derivative
The highest derivative present in the equation is \(\frac{d^2y}{dx^2}\), which is the second derivative.

### Step 3: Define the order of the differential equation
The order of a differential equation is defined as the highest derivative that appears in the equation. Since the highest derivative here is the second derivative, the order of the differential equation is 2.

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

**Order = 2**

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