Find the degree of the following :
`(1+(ds)/(dt))^((3)/(2)) = 5(d^(2)s)/(dt^(2))`

Answer

To find the degree of the given differential equation:

\[ (1 + \frac{ds}{dt})^{\frac{3}{2}} = 5 \frac{d^2s}{dt^2} \]

we will follow these steps:

### Step 1: Rewrite the Equation
First, we will eliminate the fractional exponent on the left side by cubing both sides of the equation.

\[ (1 + \frac{ds}{dt})^{\frac{3}{2}} = 5 \frac{d^2s}{dt^2} \]

Cubing both sides gives:

\[ 1 + \frac{ds}{dt} = 5^2 \left(\frac{d^2s}{dt^2}\right)^{\frac{2}{3}} \]

This simplifies to:

\[ 1 + \frac{ds}{dt} = 25 \left(\frac{d^2s}{dt^2}\right)^{\frac{2}{3}} \]

### Step 2: Identify the Derivatives
Next, we need to identify the highest order derivative present in the equation. The derivatives present are:

- \(\frac{ds}{dt}\) (first order)
- \(\frac{d^2s}{dt^2}\) (second order)

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

In our equation, the highest order derivative is \(\frac{d^2s}{dt^2}\), and it appears in the term \(25 \left(\frac{d^2s}{dt^2}\right)^{\frac{2}{3}}\).

However, since we need the degree to be defined, we must express the equation as a polynomial in terms of the derivatives.

To do this, we can rearrange the equation:

\[ (1 + \frac{ds}{dt}) = 25 \left(\frac{d^2s}{dt^2}\right)^{\frac{2}{3}} \]

This indicates that the highest order derivative \(\frac{d^2s}{dt^2}\) is raised to the power of \(\frac{2}{3}\).

Since the degree must be a whole number, we can see that the degree of the differential equation is determined by the highest integer power of the highest order derivative, which is \(2\) (from \(\frac{d^2s}{dt^2}\)).

### Final Answer
Thus, the degree of the given differential equation is:

\[ \text{Degree} = 2 \]