Determine the order and degree of the differential equation :
`((dy)/(dx))^(3)-2y(d^(2)y)/(dx^(2))=0`.

Answer

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

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

we will follow these steps:

### Step 1: Identify the derivatives present in the equation

In the given equation, we have:
- The first derivative \(\frac{dy}{dx}\)
- The second derivative \(\frac{d^{2}y}{dx^{2}}\)

### Step 2: Determine the order of the differential equation

The order of a differential equation is defined as the highest order of the derivative present in the equation.

In our case:
- The highest derivative is \(\frac{d^{2}y}{dx^{2}}\), which is a second derivative.

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

### Step 3: Determine the degree of the differential equation

The degree of a differential equation is defined as the power of the highest order derivative after the equation has been made a polynomial in derivatives.

In our equation, the highest derivative is \(\frac{d^{2}y}{dx^{2}}\) and it appears in the term \(-2y\frac{d^{2}y}{dx^{2}}\). The term \(\left(\frac{dy}{dx}\right)^{3}\) has the first derivative raised to the power of 3.

To find the degree:
- The highest derivative \(\frac{d^{2}y}{dx^{2}}\) is not raised to any power other than 1 in the equation.
- The term \(\left(\frac{dy}{dx}\right)^{3}\) is raised to the power of 3.

Since the degree is determined by the highest power of the highest order derivative, we see that the highest order derivative \(\frac{d^{2}y}{dx^{2}}\) is to the power of 1.

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

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

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