If m and n are degree and order of `(1+y_1^2)^(2//3) = y_2` , then the value of `"m+n"/"m-n"` is

Options

A. 3
B. 4
C. 5
D. 2

Answer

To solve the given problem, we need to analyze the differential equation represented by the expression \((1 + y_1^2)^{\frac{2}{3}} = y_2\), where \(y_1 = \frac{dy}{dx}\) and \(y_2 = \frac{d^2y}{dx^2}\).

### Step-by-step Solution:

1. **Identify the Variables**:
- Let \(y_1 = \frac{dy}{dx}\) (first derivative of \(y\))
- Let \(y_2 = \frac{d^2y}{dx^2}\) (second derivative of \(y\))

2. **Rewrite the Equation**:
- The given equation is \((1 + y_1^2)^{\frac{2}{3}} = y_2\).

3. **Eliminate the Rational Power**:
- To eliminate the rational exponent, we can raise both sides of the equation to the power of 3:
\[ (1 + y_1^2)^2 = y_2^3 \]

4. **Differentiate the Equation**:
- Now we need to differentiate both sides with respect to \(x\). Using the chain rule and product rule, we differentiate the left side:
\[ 2(1 + y_1^2) \cdot \frac{d}{dx}(1 + y_1^2) = 3y_2^2 \cdot \frac{dy_2}{dx} \]
- The derivative of \(1 + y_1^2\) is \(2y_1 \frac{dy_1}{dx}\), thus:
\[ 2(1 + y_1^2)(2y_1 \frac{dy_1}{dx}) = 3y_2^2 \cdot \frac{dy_2}{dx} \]

5. **Identify the Order and Degree**:
- The highest derivative present in the equation is \(y_2\) (which is the second derivative), indicating that the order \(n = 2\).
- The degree \(m\) is determined by the highest power of the highest order derivative when the equation is expressed as a polynomial. Here, the highest power of \(y_2\) is 3, so \(m = 3\).

6. **Calculate \(m+n\) and \(m-n\)**:
- Now, we can calculate \(m+n\) and \(m-n\):
\[ m + n = 3 + 2 = 5 \]
\[ m - n = 3 - 2 = 1 \]

7. **Final Calculation**:
- We need to find the value of \(\frac{m+n}{m-n}\):
\[ \frac{m+n}{m-n} = \frac{5}{1} = 5 \]

### Final Answer:
The value of \(\frac{m+n}{m-n}\) is \(5\).