The order and degree of the differential equation `(d^(2)y)/(dx^(2))=sqrt(1+((dy)/(dx))^(3))`, is

Options

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

Answer

To determine the order and degree of the given differential equation

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

we will follow these steps:

### Step 1: Identify the order of the differential equation
The order of a differential equation is determined by the highest derivative present in the equation. In our case, we have:

\[ \frac{d^2y}{dx^2} \]

This is the second derivative of \(y\) with respect to \(x\). Therefore, the order of the differential equation is 2.

### Step 2: Rearranging the equation to find the degree
The degree of a differential equation is defined as the power of the highest order derivative after the equation has been made a polynomial in derivatives.

Let's square both sides of the equation to eliminate the square root:

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

### Step 3: Identify the highest order derivative and its power
Now, we can see that the highest order derivative is still \(\frac{d^2y}{dx^2}\) and it appears squared on the left side of the equation.

Thus, the degree of the differential equation is 2 (since the power of the highest order derivative, \(\frac{d^2y}{dx^2}\), is 2).

### Conclusion
From our analysis, we find that:
- The order of the differential equation is **2**.
- The degree of the differential equation is **2**.

Thus, the answer is:

**Order: 2, Degree: 2**