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

Options

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

Answer

To find the degree of the given differential equation

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

we will follow these steps:

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

**Hint:** Look for the derivative with the highest order in the equation.

### Step 2: Rearrange the equation
To analyze the degree, we can rearrange the equation. The left-hand side is raised to a power, so we can square both sides to eliminate the exponent.

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

**Hint:** When dealing with powers, consider manipulating the equation to simplify the terms.

### Step 3: Simplify the equation
Now, we can express the left-hand side without the exponent:

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

**Hint:** Focus on isolating the highest order derivative to clearly see its power.

### Step 4: Determine the degree
In the original equation, the highest order derivative is \(\frac{d^{2}y}{dx^{2}}\), and it appears to the power of 1 in the rearranged equation. The degree of a differential equation is defined as the power of the highest order derivative when the equation is a polynomial in derivatives.

Since the highest order derivative \(\frac{d^{2}y}{dx^{2}}\) is raised to the power of 1, the degree of the differential equation is 2.

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

### Conclusion
Thus, the degree of the differential equation is 2.

**Final Answer:** The degree of the differential equation is 2 (Option C).