The general solution of the differential equation `(x^(2) + x + 1) dy + (y^(2) + y + 1) dx = 0` is (x + y + 1) = A (1 + Bx + Cy + Dxy) , where B , C and D are constants and A is parameter
What is C equal to ?

Options

A. (A) 1
B. (B) `-1`
C. (C) 2
D. (D) None of these

Answer

To solve the differential equation \((x^{2} + x + 1) dy + (y^{2} + y + 1) dx = 0\) and find the value of \(C\) in the general solution given by \((x + y + 1) = A(1 + Bx + Cy + Dxy)\), we will follow these steps:

### Step 1: Rewrite the Differential Equation
We start with the given differential equation:
\[ (x^{2} + x + 1) dy + (y^{2} + y + 1) dx = 0 \]
We can rearrange this to separate the variables:
\[ \frac{dy}{y^{2} + y + 1} = -\frac{dx}{x^{2} + x + 1} \]

### Step 2: Integrate Both Sides
Next, we integrate both sides:
\[ \int \frac{dy}{y^{2} + y + 1} = -\int \frac{dx}{x^{2} + x + 1} \]

### Step 3: Completing the Square
For the integration, we need to complete the square for both the numerator and denominator:

- For \(y^{2} + y + 1\):
\[ y^{2} + y + 1 = \left(y + \frac{1}{2}\right)^{2} + \frac{3}{4} \]

- For \(x^{2} + x + 1\):
\[ x^{2} + x + 1 = \left(x + \frac{1}{2}\right)^{2} + \frac{3}{4} \]

### Step 4: Use the Integration Formula
Using the formula \(\int \frac{dx}{x^{2} + a^{2}} = \frac{1}{a} \tan^{-1} \left(\frac{x}{a}\right) + C\), we can integrate:
\[ \int \frac{dy}{\left(y + \frac{1}{2}\right)^{2} + \frac{3}{4}} = \frac{2}{\sqrt{3}} \tan^{-1}\left(\frac{2y + 1}{\sqrt{3}}\right) + C_1 \]
\[ -\int \frac{dx}{\left(x + \frac{1}{2}\right)^{2} + \frac{3}{4}} = -\frac{2}{\sqrt{3}} \tan^{-1}\left(\frac{2x + 1}{\sqrt{3}}\right) + C_2 \]

### Step 5: Combine the Results
Combining the results from the integrations gives:
\[ \frac{2}{\sqrt{3}} \tan^{-1}\left(\frac{2y + 1}{\sqrt{3}}\right) + \frac{2}{\sqrt{3}} \tan^{-1}\left(\frac{2x + 1}{\sqrt{3}}\right) = C \]

### Step 6: Express in the Required Form
We need to express this in the form:
\[ x + y + 1 = A(1 + Bx + Cy + Dxy) \]
From the integration results, we can derive relationships between \(A\), \(B\), \(C\), and \(D\).

### Step 7: Identify the Value of \(C\)
By comparing coefficients in the derived expression with the required form, we can identify \(C\).

After performing the necessary algebraic manipulations and comparisons, we find that:
\[ C = -1 \]

### Final Answer
Thus, the value of \(C\) is:
\[ \boxed{-1} \]