STATEMENT-1 : If m and n are respectively order and degree of differential equation , then m and n are mutually independent.
STATEMENT-2 : If general solution of a differential equation contains two arbitrary constants, then its order is 2.
STATEMENT-3 : The order and degree of differential equation `sqrt(1+((dy)/(dx))^(2)) = (x(d^(2)y)/(dx^(2)))^(1//3)` are 2 and 2 respectively.

Options

A. (a)T F T
B. (b)T T T
C. (c)F F F
D. (d)F F T

Answer

To solve the problem, we need to analyze each statement regarding differential equations and determine their validity.

### Step-by-Step Solution:

**Statement 1:** "If m and n are respectively order and degree of differential equation, then m and n are mutually independent."

- **Explanation:** The order of a differential equation (m) is the highest derivative present in the equation, while the degree (n) is the power of the highest derivative when the equation is a polynomial in derivatives. These two properties are generally independent of each other; knowing one does not provide information about the other. Thus, this statement is **True**.

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**Statement 2:** "If the general solution of a differential equation contains two arbitrary constants, then its order is 2."

- **Explanation:** The number of arbitrary constants in the general solution of a differential equation corresponds to its order. If a differential equation has a general solution that includes two arbitrary constants, it indicates that the order of the differential equation is 2. Therefore, this statement is also **True**.

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**Statement 3:** "The order and degree of the differential equation \( \sqrt{1 + \left(\frac{dy}{dx}\right)^2} = \left(x \frac{d^2y}{dx^2}\right)^{\frac{1}{3}} \) are 2 and 2 respectively."

- **Explanation:**
1. Start with the given equation:
\[ \sqrt{1 + \left(\frac{dy}{dx}\right)^2} = \left(x \frac{d^2y}{dx^2}\right)^{\frac{1}{3}} \]
2. Square both sides to eliminate the square root:
\[ 1 + \left(\frac{dy}{dx}\right)^2 = \left(x \frac{d^2y}{dx^2}\right)^{\frac{2}{3}} \]
3. Rearranging gives us:
\[ \left(\frac{dy}{dx}\right)^2 = \left(x \frac{d^2y}{dx^2}\right)^{\frac{2}{3}} - 1 \]
4. The highest derivative present is \( \frac{d^2y}{dx^2} \), indicating that the order is 2.
5. To find the degree, we observe that the equation can be expressed in polynomial form with respect to the derivatives, so the degree is also 2.

Thus, the order is 2 and the degree is 2, making this statement **True**.

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### Conclusion:
All three statements are true. Therefore, the correct option is **B: All statements are true**.