Which of the following statements on ordinary differential equations is/are true ?
(i) The number of arbitrary constants is same as the degree of the differential equation.
(ii) A linear differential equation can contain products of the dependent variable and its derivatives.
(iii) A particular integral cannot contains arbitrary constants.
(iv) By putting `v=(y)/(x)` any homogeneous first order differential equation transforms to variable separable form.

Options

A. (i) and (iii) only
B. (ii) and (iii) only
C. (iii) only
D. (i) and (iv) only

Explanation

To determine the truth of the statements regarding ordinary differential equations, we will analyze each statement step by step.

### Step-by-Step Solution:

1. **Statement (i)**: "The number of arbitrary constants is the same as the degree of the differential equation."
- **Analysis**: This statement is **false**. The number of arbitrary constants in the general solution of a differential equation is equal to the order of the differential equation, not the degree. The degree refers to the highest power of the highest derivative in the equation, which does not necessarily correlate with the number of arbitrary constants.
- **Conclusion**: **False**

2. **Statement (ii)**: "A linear differential equation can contain products of the dependent variable and its derivatives."
- **Analysis**: This statement is also **false**. A linear differential equation is defined as one in which the dependent variable and its derivatives appear linearly, meaning they cannot be multiplied together. If they are multiplied, the equation becomes nonlinear.
- **Conclusion**: **False**

3. **Statement (iii)**: "A particular integral cannot contain arbitrary constants."
- **Analysis**: This statement is **true**. A particular integral is a specific solution to a non-homogeneous differential equation and does not include arbitrary constants. It is determined by the non-homogeneous part of the equation.
- **Conclusion**: **True**

4. **Statement (iv)**: "By putting v = y/x, any homogeneous first-order differential equation transforms to variable separable form."
- **Analysis**: This statement is **false**. While the substitution \( v = \frac{y}{x} \) can sometimes help in solving homogeneous first-order differential equations, it does not universally guarantee that the equation will transform into a separable form. It depends on the specific structure of the equation.
- **Conclusion**: **False**

### Final Summary of Statements:
- (i) False
- (ii) False
- (iii) True
- (iv) False