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 by rewriting the equation in the form of \( \frac{dy}{dx} \):
\[ (3x + 3y - 4) dy = -(x + y) dx \]
Dividing both sides by \( dx \) and \( (3x + 3y - 4) \):
\[ \frac{dy}{dx} = -\frac{x + y}{3x + 3y - 4} \]

### Step 2: Simplify the Right Side
We can simplify the right side:
\[ \frac{dy}{dx} = -\frac{x + y}{3(x + y) - 4} \]
Let \( v = x + y \). Thus, we can rewrite the equation as:
\[ \frac{dy}{dx} = -\frac{v}{3v - 4} \]

### Step 3: Express dy/dx in terms of dv/dx
Using the substitution \( v = x + y \), we differentiate:
\[ \frac{dv}{dx} = 1 + \frac{dy}{dx} \]
Substituting \( \frac{dy}{dx} \) into this gives:
\[ \frac{dv}{dx} = 1 - \frac{v}{3v - 4} \]
Rearranging this, we have:
\[ \frac{dv}{dx} = \frac{(3v - 4) - v}{3v - 4} = \frac{2v - 4}{3v - 4} \]

### Step 4: Separate Variables
Now, we separate the variables:
\[ \frac{3v - 4}{2v - 4} dv = dx \]

### Step 5: Integrate Both Sides
Integrate both sides:
\[ \int \frac{3v - 4}{2v - 4} dv = \int dx \]
We can simplify the left-hand side:
\[ \int \left( \frac{3}{2} + \frac{2}{2v - 4} \right) dv = x + C \]
This gives:
\[ \frac{3}{2}v + \log|2v - 4| = x + C \]

### Step 6: Substitute Back for v
Substituting back \( v = x + y \):
\[ \frac{3}{2}(x + y) + \log|2(x + y) - 4| = x + C \]

### Final Step: Rearranging
Rearranging gives us the implicit solution of the differential equation.

### Summary
The solution of the differential equation \( (3x + 3y - 4) dy + (x + y) dx = 0 \) is given by:
\[ \frac{3}{2}(x + y) + \log|2(x + y) - 4| = x + C \]