Consider the following statements :
1 . The general solution of `(dy)/(dx) = f(x) +x ` is of the form y = g(x) + C , where C is an arbitrary constant .
II . The degree of `((dy)/(dx))^2 = f(x)` is 2 .
Which of the above statements is /are correct ?

Options

A. A. Only I
B. B. Only II
C. C. Both I and II
D. D. Neither I nor II

Answer

To solve the given problem, we need to analyze the two statements provided.

### Step 1: Analyze the first statement
The first statement claims that the general solution of the differential equation

\[ \frac{dy}{dx} = f(x) + x \]

is of the form

\[ y = g(x) + C \]

where \( C \) is an arbitrary constant.

**Solution:**
1. We start with the equation:

\[ \frac{dy}{dx} = f(x) + x \]

2. Rearranging gives:

\[ dy = (f(x) + x) \, dx \]

3. Now, we integrate both sides:

\[ \int dy = \int (f(x) + x) \, dx \]

4. This leads to:

\[ y = \int f(x) \, dx + \int x \, dx + C \]

5. The integral of \( x \) is:

\[ \int x \, dx = \frac{x^2}{2} \]

6. Therefore, we can write:

\[ y = \int f(x) \, dx + \frac{x^2}{2} + C \]

7. We can denote \( \int f(x) \, dx \) as \( g(x) \), thus:

\[ y = g(x) + \frac{x^2}{2} + C \]

8. This shows that the general solution is not simply \( y = g(x) + C \) but includes the term \( \frac{x^2}{2} \).

**Conclusion for Statement 1:**
The first statement is **incorrect** because the general solution is not of the form \( y = g(x) + C \).

### Step 2: Analyze the second statement
The second statement claims that the degree of

\[ \left(\frac{dy}{dx}\right)^2 = f(x) \]

is 2.

**Solution:**
1. The degree of a differential equation is defined as the power of the highest derivative in the equation.
2. In the equation \( \left(\frac{dy}{dx}\right)^2 = f(x) \), the highest derivative is \( \frac{dy}{dx} \) raised to the power of 2.
3. Therefore, the degree of this equation is indeed 2.

**Conclusion for Statement 2:**
The second statement is **correct**.

### Final Conclusion
- Statement I is incorrect.
- Statement II is correct.

Thus, the correct answer is that only the second statement is true.