The degree of the differential equation `[1+((dy)/(dx))^(2)]^(3//2)=(d^(2)y)/(dx^(2))"is"`

Options

A. 4
B. `(3)/(2)`
C. not defined
D. 2

Explanation

To find the degree of the given differential equation

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

we will follow these steps:

### Step 1: Understand the definition of degree
The degree of a differential equation is defined as the highest power of the highest order derivative present in the equation when it is expressed as a polynomial equation in derivatives.

### Step 2: Rewrite the equation
The given equation is

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

To simplify this, we can eliminate the fractional exponent by squaring both sides.

### Step 3: Square both sides
Squaring both sides gives:

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

### Step 4: Identify the derivatives
In the equation

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

we have two derivatives:
- The first derivative \(\frac{dy}{dx}\) appears in the term \(\left(\frac{dy}{dx}\right)^2\).
- The second derivative \(\frac{d^2y}{dx^2}\) appears in the term \(\left(\frac{d^2y}{dx^2}\right)^2\).

### Step 5: Determine the highest order derivative
The highest order derivative in this equation is \(\frac{d^2y}{dx^2}\), which is a second-order derivative.

### Step 6: Determine the highest power of the highest order derivative
The highest power of the highest order derivative \(\frac{d^2y}{dx^2}\) in the equation is 2, as seen in the term \(\left(\frac{d^2y}{dx^2}\right)^2\).

### Conclusion
Thus, the degree of the given differential equation is

\[ \text{Degree} = 2 \]