If m and n denote respectively the order and degree of a differential equation
`[a+((dy)/(dx))^(6)]^(7//5)=b(d^(2)y)/(dx^(2))` then the value of (m, n) will be

Options

A. `(1, 7)`
B. `(1, 6)`
C. `(2, 5)`
D. `(2, 6)`

Answer

To find the order (m) and degree (n) of the given differential equation

\[ \left(a + \left(\frac{dy}{dx}\right)^6\right)^{\frac{7}{5}} = b \frac{d^2y}{dx^2} \]

we will follow these steps:

### Step 1: Clear the Fractional Power
To eliminate the fractional exponent, we can raise both sides of the equation to the power of 5:

\[ \left(\left(a + \left(\frac{dy}{dx}\right)^6\right)^{\frac{7}{5}}\right)^5 = \left(b \frac{d^2y}{dx^2}\right)^5 \]

This simplifies to:

\[ a + \left(\frac{dy}{dx}\right)^6 = b^5 \left(\frac{d^2y}{dx^2}\right)^5 \]

### Step 2: 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 equation, the highest derivative is \(\frac{d^2y}{dx^2}\), which is a second derivative. Therefore, the order \(m\) is:

\[ m = 2 \]

### Step 3: Identify 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 case, the highest order derivative \(\frac{d^2y}{dx^2}\) is raised to the power of 5. Therefore, the degree \(n\) is:

\[ n = 5 \]

### Conclusion
Thus, the values of \(m\) and \(n\) are:

\[ (m, n) = (2, 5) \]

---