The order and degree of the differential equation `[1+((dy)/(dx))^(2)]^(3//4)=((d^(2)y)/(dx^(2)))^(1//3)`

Options

A. (2,4)
B. (2,5)
C. (2,1)
D. None of these

Answer

To find the order and degree of the differential equation

\[ \left[1+\left(\frac{dy}{dx}\right)^2\right]^{\frac{3}{4}}=\left(\frac{d^2y}{dx^2}\right)^{\frac{1}{3}}, \]

we will follow these steps:

### Step 1: Identify the highest derivative
The order of a differential equation is determined by the highest order derivative present in the equation. Here, we have two derivatives:

1. \(\frac{dy}{dx}\) (first derivative)
2. \(\frac{d^2y}{dx^2}\) (second derivative)

The highest derivative is \(\frac{d^2y}{dx^2}\), which is of order 2.

**Hint:** Look for the derivative with the highest order to determine the order of the differential equation.

### Step 2: Determine the degree of the differential 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.

In our equation, we have:

\[ \left[1+\left(\frac{dy}{dx}\right)^2\right]^{\frac{3}{4}}=\left(\frac{d^2y}{dx^2}\right)^{\frac{1}{3}}. \]

To find the degree, we need to eliminate the fractional powers. We can do this by raising both sides of the equation to suitable powers.

### Step 3: Eliminate the fractional powers
First, we can raise both sides to the power of \(4\) to eliminate the \(\frac{3}{4}\) on the left side:

\[ \left[1+\left(\frac{dy}{dx}\right)^2\right]^3 = \left(\frac{d^2y}{dx^2}\right)^{\frac{4}{3}}. \]

Next, we raise both sides to the power of \(3\) to eliminate the \(\frac{1}{3}\) on the right side:

\[ \left[1+\left(\frac{dy}{dx}\right)^2\right]^9 = \left(\frac{d^2y}{dx^2}\right)^4. \]

### Step 4: Identify the degree
Now, we can see that the equation is now in polynomial form. The highest power of the highest order derivative \(\frac{d^2y}{dx^2}\) is \(4\). Therefore, the degree of the differential equation is \(4\).

### Conclusion
Thus, the order of the differential equation is \(2\) and the degree is \(4\).

**Final Answer:**
- Order: \(2\)
- Degree: \(4\)