Write the order and degree of the following differential equations :
`((ds)/(dx))^(4)+3s(d^(2)s)/(dt^(2))=0`

Answer

To determine the order and degree of the given differential equation:

\[ \left(\frac{ds}{dx}\right)^{4} + 3s \frac{d^{2}s}{dt^{2}} = 0 \]

we will follow these steps:

### Step 1: Identify the Derivatives
First, we need to identify all the derivatives present in the equation.

- The term \(\frac{ds}{dx}\) is the first derivative of \(s\) with respect to \(x\).
- The term \(\frac{d^{2}s}{dt^{2}}\) is the second derivative of \(s\) with respect to \(t\).

### Step 2: Determine the Order
The order of a differential equation is defined as the highest order of derivative present in the equation.

- The highest derivative in the equation is \(\frac{d^{2}s}{dt^{2}}\), which is a second derivative.

Thus, the **order** of the differential equation is **2**.

### Step 3: Determine the Degree
The degree of a differential equation is defined as the power of the highest order derivative when the equation is a polynomial in derivatives.

- The highest derivative \(\frac{d^{2}s}{dt^{2}}\) appears in the equation, and its power is \(1\) (since it is not raised to any power).
- The term \(\left(\frac{ds}{dx}\right)^{4}\) is also a polynomial term, but it does not affect the degree of the highest derivative.

Thus, the **degree** of the differential equation is **1**.

### Final Answer
- **Order**: 2
- **Degree**: 1