The solution of `( 3x + 3y -4) dy + (x+y) dx =0` is given by :

Options

A. `(x+y) + bg |x-y|-4 |=C`
B. `3x 3y - 4 log |x-4|=C`
C. `(3)/(2) (x+y) + bg |x+y -2 |x=c`
D. None of these

Answer

To solve the differential equation \( (3x + 3y - 4) dy + (x + y) dx = 0 \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:
\[ (3x + 3y - 4) dy + (x + y) dx = 0 \]
We can rearrange this to isolate \( dy \) and \( dx \):
\[ (3x + 3y - 4) dy = -(x + y) dx \]
Dividing both sides by \( dx \) gives:
\[ dy = -\frac{x + y}{3x + 3y - 4} dx \]

### Step 2: Find \( \frac{dy}{dx} \)
From the equation above, we can express \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = -\frac{x + y}{3x + 3y - 4} \]

### Step 3: Substitute \( x + y = v \)
Let \( v = x + y \). Then, we differentiate \( v \) with respect to \( x \):
\[ \frac{dv}{dx} = 1 + \frac{dy}{dx} \]
This implies:
\[ \frac{dy}{dx} = \frac{dv}{dx} - 1 \]

### Step 4: Substitute back into the equation
Substituting \( \frac{dy}{dx} \) into our expression gives:
\[ \frac{dv}{dx} - 1 = -\frac{v}{3v - 4} \]
Rearranging this, we get:
\[ \frac{dv}{dx} = -\frac{v}{3v - 4} + 1 \]

### Step 5: Simplify the right-hand side
Combining the terms on the right-hand side:
\[ \frac{dv}{dx} = 1 - \frac{v}{3v - 4} = \frac{(3v - 4) - v}{3v - 4} = \frac{2v - 4}{3v - 4} \]

### Step 6: Separate variables
We can separate the variables:
\[ \frac{3v - 4}{2v - 4} dv = dx \]

### Step 7: Integrate both sides
Now we integrate both sides:
\[ \int \frac{3v - 4}{2v - 4} dv = \int dx \]
This can be simplified further by breaking it down into simpler fractions.

### Step 8: Solve the integral
Let’s perform the integration:
1. Rewrite \( \frac{3v - 4}{2v - 4} \) as \( \frac{3}{2} + \frac{2}{2v - 4} \).
2. Integrate:
\[ \int \left( \frac{3}{2} + \frac{2}{2v - 4} \right) dv = \frac{3}{2}v + \log|2v - 4| + C \]
3. The right side integrates to \( x + C \).

### Step 9: Substitute back for \( v \)
Substituting back \( v = x + y \):
\[ \frac{3}{2}(x + y) + \log|2(x + y) - 4| = x + C \]

### Final Step: Rearranging the equation
Rearranging gives us the implicit solution to the differential equation:
\[ \frac{3}{2}(x + y) + \log|2(x + y) - 4| = x + C \]