Write the sum of the order and degree of the differential equation : `d/(dx){((dy)/(dx))^4}=0`

Answer

To solve the problem, we need to find the sum of the order and degree of the given differential equation:

\[ \frac{d}{dx}\left(\left(\frac{dy}{dx}\right)^4\right) = 0 \]

### Step 1: Identify the highest order derivative
The first step is to identify the highest order derivative present in the equation.

In the given equation, we have:
\[ \frac{dy}{dx} \]
This is the first derivative of \(y\) with respect to \(x\). However, since it is raised to the power of 4, we need to differentiate it.

### Step 2: Differentiate using the chain rule
To differentiate \(\left(\frac{dy}{dx}\right)^4\), we apply the chain rule:
\[ \frac{d}{dx}\left(\left(\frac{dy}{dx}\right)^4\right) = 4\left(\frac{dy}{dx}\right)^3 \cdot \frac{d^2y}{dx^2} \]
This shows that the highest derivative present after differentiation is \(\frac{d^2y}{dx^2}\), which is the second derivative.

### Step 3: Determine the order of the differential equation
The order of a differential equation is defined as the highest derivative present in the equation. From our differentiation, we see that the highest derivative is \(\frac{d^2y}{dx^2}\), which is of order 2.

### Step 4: Determine the degree of the differential equation
The degree of a differential equation is defined as the power of the highest order derivative when the equation is a polynomial in derivatives. In our case, the highest order derivative is \(\frac{d^2y}{dx^2}\) and it appears to the power of 1 (since we have \(4\left(\frac{dy}{dx}\right)^3 \cdot \frac{d^2y}{dx^2} = 0\)).

Thus, the degree is 1.

### Step 5: Calculate the sum of the order and degree
Now that we have both the order and degree:
- Order = 2
- Degree = 1

The sum of the order and degree is:
\[ \text{Sum} = \text{Order} + \text{Degree} = 2 + 1 = 3 \]

### Final Answer
The sum of the order and degree of the differential equation is **3**.

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