What is the degree of the differential equation , `k "" (d^(2) y)/(dx^(2)) = [ 1 + ((dy)/(dx))^(3)]^(3//2)` , where k is a constant ?

Options

A. 1
B. 2
C. 3
D. 4

Answer

To determine the degree of the given differential equation

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

we will follow these steps:

### Step 1: Identify the highest order derivative
The highest order derivative in the equation is \(\frac{d^2y}{dx^2}\), which is the second derivative of \(y\) with respect to \(x\).

### Step 2: Rewrite the equation
We can rearrange the equation to express it in a standard form:

\[ k \frac{d^2y}{dx^2} - \left[ 1 + \left( \frac{dy}{dx} \right)^3 \right]^{\frac{3}{2}} = 0. \]

### Step 3: 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 is \(\frac{d^2y}{dx^2}\) and it appears to the first power (i.e., it is not raised to any power greater than 1).

### Conclusion
Thus, the degree of the differential equation is 1.

The final answer is:

**Degree = 1**

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