The degree of the differential equation `[1 + ((dy)/(dx))^(2)]^(5//3) = (d^(2)y)/(dx^(2))` is

Options

A. 1
B. 5
C. `10//3`
D. 3

Answer

To find the degree of the differential equation given by

\[ \left[1 + \left(\frac{dy}{dx}\right)^2\right]^{\frac{5}{3}} = \frac{d^2y}{dx^2} \]

we will follow these steps:

### Step 1: Eliminate the Fractional Power
To eliminate the fractional power, we can raise both sides of the equation to the power of 3. This gives us:

\[ \left(1 + \left(\frac{dy}{dx}\right)^2\right)^5 = \left(\frac{d^2y}{dx^2}\right)^3 \]

### Step 2: Identify the Highest Order Derivative
Next, we need to identify the highest order derivative in the equation. The left side contains the first derivative \(\frac{dy}{dx}\) and the right side contains the second derivative \(\frac{d^2y}{dx^2}\).

### Step 3: Determine the Degree of the Highest Order Derivative
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 derivatives.

In our case, the highest order derivative is \(\frac{d^2y}{dx^2}\) and it appears raised to the power of 3. Thus, the degree of the differential equation is 3.

### Final Answer
Therefore, the degree of the differential equation is

\[ \text{Degree} = 3 \]

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