Find the order and degree (if defined) of the following differential equations:
`(d^(2)y)/(dx^(2)) = {1+((dy)/(dx))^(4)}^(5//3)`

Explanation

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

\[ \frac{d^2y}{dx^2} = \left(1 + \left(\frac{dy}{dx}\right)^4\right)^{\frac{5}{3}} \]

we will follow these steps:

### Step 1: Identify the highest derivative
The left-hand side of the equation contains the term \(\frac{d^2y}{dx^2}\), which is the second derivative of \(y\).

### Step 2: Determine the order
The order of a differential equation is defined as the highest derivative present in the equation. In this case, the highest derivative is \(\frac{d^2y}{dx^2}\), which is the second derivative.

Thus, the order of the differential equation is:
\[ \text{Order} = 2 \]

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

In our equation, the highest derivative \(\frac{d^2y}{dx^2}\) is raised to the power of 1 (it appears as is), but the right-hand side contains the term \(\left(1 + \left(\frac{dy}{dx}\right)^4\right)^{\frac{5}{3}}\).

To find the degree, we need to express the equation in a polynomial form. We can rewrite the equation by eliminating the fractional exponent.

### Step 4: Eliminate the fractional exponent
We can multiply both sides of the equation by \(3\) to eliminate the fractional exponent:
\[ \left(\frac{d^2y}{dx^2}\right)^3 = 1 + \left(\frac{dy}{dx}\right)^4 \]

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

### Step 5: Determine the degree
Now, since the highest derivative \(\frac{d^2y}{dx^2}\) is raised to the power of 3, we can conclude that the degree of the differential equation is:
\[ \text{Degree} = 3 \]

### Final Result
Therefore, for the given differential equation:
- **Order**: 2
- **Degree**: 3