solution of ` ( x+ y-1) dx + (2x+ 2y -3) dy=0` is :

Options

A. `y+x + log ( x + y-2)=c`
B. ` y+ 2x + log (x + y -2 ) = c`
C. `2y + x + log ( x + y-2)=C`
D. `2y + 2x +log ( x+ y-2) =C`

Answer

To solve the differential equation \( (x + y - 1) \, dx + (2x + 2y - 3) \, dy = 0 \), we can follow these steps:

### Step 1: Rewrite the Equation
We start by rewriting the equation in the form of \(\frac{dy}{dx}\):
\[ (2x + 2y - 3) \, dy = -(x + y - 1) \, dx \]
Dividing both sides by \(dx\) and rearranging gives:
\[ \frac{dy}{dx} = -\frac{x + y - 1}{2x + 2y - 3} \]

### Step 2: Substitute \(v = x + y\)
Let \(v = x + y\). Then, differentiating both sides gives:
\[ \frac{dv}{dx} = 1 + \frac{dy}{dx} \]
Thus, we can express \(\frac{dy}{dx}\) as:
\[ \frac{dy}{dx} = \frac{dv}{dx} - 1 \]

### Step 3: Substitute into the Equation
Substituting \(\frac{dy}{dx}\) into the equation gives:
\[ \frac{dv}{dx} - 1 = -\frac{v - 1}{2v - 3} \]
Rearranging this gives:
\[ \frac{dv}{dx} = -\frac{v - 1}{2v - 3} + 1 \]

### Step 4: Simplify the Right Side
To simplify the right side:
\[ \frac{dv}{dx} = 1 - \frac{v - 1}{2v - 3} = \frac{(2v - 3) - (v - 1)}{2v - 3} = \frac{v - 2}{2v - 3} \]

### Step 5: Separate Variables
Now we can separate the variables:
\[ \frac{2v - 3}{v - 2} \, dv = dx \]

### Step 6: Integrate Both Sides
Integrating both sides:
\[ \int \frac{2v - 3}{v - 2} \, dv = \int dx \]
We can simplify the left side:
\[ \int \left(2 + \frac{1}{v - 2}\right) \, dv = x + C \]
This gives:
\[ 2v + \log|v - 2| = x + C \]

### Step 7: Substitute Back for \(v\)
Substituting back \(v = x + y\):
\[ 2(x + y) + \log|x + y - 2| = x + C \]

### Step 8: Rearranging the Equation
Rearranging gives:
\[ x + 2y + \log|x + y - 2| = C \]

### Final Solution
Thus, the solution to the differential equation is:
\[ x + 2y + \log|x + y - 2| = C \]