`(d^(4)y)/(dx^(4))=[1+((dy)/(dx))^(2)]^(3//2)` . Find order and degree of differential equation.

Options

A. 4, 6
B. 4, 3
C. 2, 4
D. 4, 2

Answer

To find the order and degree of the given differential equation

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

we will follow these steps:

### Step 1: Identify the highest order derivative
The left-hand side of the equation contains the term \(\frac{d^4y}{dx^4}\), which is the fourth derivative of \(y\) with respect to \(x\).

### Step 2: Determine the order of the differential equation
The order of a differential equation is defined as the highest order of derivative present in the equation. Since the highest derivative here is \(\frac{d^4y}{dx^4}\), the order of the differential equation is:

\[ \text{Order} = 4. \]

### Step 3: Identify 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 expressed as a polynomial in terms of the highest order derivative.

To find the degree, we first need to express the equation in a suitable form. The right-hand side of the equation contains a fractional exponent. To eliminate this, we can square both sides of the equation:

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

### Step 4: Analyze the new equation
Now we have:

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

In this equation, the highest order derivative \(\frac{d^4y}{dx^4}\) is raised to the power of 2.

### Step 5: Determine the degree
Thus, the degree of the differential equation is:

\[ \text{Degree} = 2. \]

### Final Answer
- **Order**: 4
- **Degree**: 2