The sum of order and degree of the differential equation `1 + ((d^(4)y)/(dx^(4)))^(3)= root(3)(1- (dy)/(dx))` is

Options

A. 9
B. 4
C. 13
D. 5

Answer

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

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

### Step 1: Identify the highest derivative (Order)
The order of a differential equation is defined as the highest derivative present in the equation. In this case, we have:

- The highest derivative is \(\frac{d^4y}{dx^4}\).

Thus, the order of the differential equation is **4**.

### Step 2: Rearranging the equation to find the Degree
The degree of a differential equation is defined as the power of the highest derivative after the equation has been arranged in a polynomial form (with respect to the highest derivative).

First, we need to eliminate the cube root and the cube on the left side. To do this, we will cube both sides of the equation:

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

This simplifies to:

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

Now we can isolate the term involving the highest derivative:

\[ \left(\frac{d^4y}{dx^4}\right)^3 = -\frac{dy}{dx} \]

### Step 3: Determine the Degree
Now, we can see that the term \(\left(\frac{d^4y}{dx^4}\right)^3\) is raised to the power of 3. Therefore, the degree of the differential equation is **3**.

### Step 4: Calculate the Sum of Order and Degree
Finally, we calculate the sum of the order and degree:

\[ \text{Sum} = \text{Order} + \text{Degree} = 4 + 3 = 7 \]

### Final Answer
The sum of the order and degree of the differential equation is **7**.

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