Write the order and degree of the following differential equations.
`sqrt(1+(dy)/(dx))= ((d^2y)/(dx^2))^(1/3)`

Answer

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

\[ \sqrt{1 + \frac{dy}{dx}} = \left(\frac{d^2y}{dx^2}\right)^{1/3} \]

we will follow these steps:

### Step 1: Identify the highest order derivative
In the given equation, we have two derivatives:
- \(\frac{dy}{dx}\) (first order derivative)
- \(\frac{d^2y}{dx^2}\) (second order derivative)

The highest order derivative here is \(\frac{d^2y}{dx^2}\), which is of order 2.

**Hint for Step 1:** Look for the derivative with the highest order in the equation to determine the order.

### Step 2: Determine the degree of the highest order derivative
The degree of a differential equation is defined as the power of the highest order derivative when the equation is expressed in a polynomial form in terms of the derivatives.

In our case, the highest order derivative is \(\frac{d^2y}{dx^2}\) raised to the power of \(\frac{1}{3}\). Since the degree must be a positive integer, we cannot directly take \(\frac{1}{3}\) as the degree.

**Hint for Step 2:** If the power of the highest order derivative is not a positive integer, manipulate the equation to eliminate fractional powers.

### Step 3: Eliminate the fractional power
To eliminate the fractional power, we can cube both sides of the equation:

\[ \left(\sqrt{1 + \frac{dy}{dx}}\right)^3 = \frac{d^2y}{dx^2} \]

This simplifies to:

\[ 1 + \frac{dy}{dx} = \left(\frac{d^2y}{dx^2}\right)^3 \]

Now we can see that the highest order derivative \(\frac{d^2y}{dx^2}\) is raised to the power of 3.

**Hint for Step 3:** Use algebraic manipulation to express the equation in a form where the derivatives are raised to integer powers.

### Step 4: Identify the new degree
Now, we can rewrite the equation as:

\[ \left(\frac{d^2y}{dx^2}\right)^3 - \left(1 + \frac{dy}{dx}\right) = 0 \]

In this form, the highest order derivative \(\frac{d^2y}{dx^2}\) is raised to the power of 3.

Thus, the degree of the differential equation is 3.

### Conclusion
The order of the differential equation is 2, and the degree is 3.

**Final Answer:**
- Order: 2
- Degree: 3