The order and degree of the differential equation `((dy)/(dx)) ^(1//3) -4 (d ^(2)y)/(dx ^(2)) -7x=0` are `alpha and beta,` then the value of `(alpha +beta)` is:

Options

A. 3
B. 4
C. 2
D. 5

Answer

To solve the problem of finding the order and degree of the given differential equation

\[ \left(\frac{dy}{dx}\right)^{\frac{1}{3}} - 4 \frac{d^2y}{dx^2} - 7x = 0, \]

we will follow these steps:

### Step 1: Identify the highest order derivative
The given equation contains the first derivative \(\frac{dy}{dx}\) and the second derivative \(\frac{d^2y}{dx^2}\). The highest order derivative present is \(\frac{d^2y}{dx^2}\), which indicates that the order of the differential equation is 2.

**Hint:** The order of a differential equation is determined by the highest derivative present in the equation.

### Step 2: Rewrite the equation for clarity
To analyze the degree, we need to express the equation in a form where all derivatives are raised to integer powers. The original equation can be rewritten as:

\[ \left(\frac{dy}{dx}\right)^{\frac{1}{3}} = 4 \frac{d^2y}{dx^2} + 7x. \]

**Hint:** Isolate the term with the highest derivative to facilitate the next steps.

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

\[ \left(\frac{dy}{dx}\right) = \left(4 \frac{d^2y}{dx^2} + 7x\right)^3. \]

**Hint:** Cubing both sides helps to convert the fractional power into an integer power.

### Step 4: Analyze the resulting equation
After cubing, we have:

\[ \frac{dy}{dx} = \left(4 \frac{d^2y}{dx^2} + 7x\right)^3. \]

In this equation, the highest derivative is still \(\frac{d^2y}{dx^2}\), and it appears in a cubic term.

### Step 5: Determine the degree
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 this case, the highest order derivative \(\frac{d^2y}{dx^2}\) appears to the first power inside the cubic expression, which means the degree is 3.

**Hint:** The degree is determined by the power of the highest order derivative in the polynomial form of the equation.

### Step 6: Calculate \(\alpha + \beta\)
Now that we have determined:
- Order \(\alpha = 2\)
- Degree \(\beta = 3\)

We can find the sum:

\[ \alpha + \beta = 2 + 3 = 5. \]

### Final Answer
Thus, the value of \((\alpha + \beta)\) is:

\[ \boxed{5}. \]