Solution of the differential equation `(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 will follow these steps:

### Step 1: Rewrite the Equation
We start with the given differential equation:
\[ (x+y-1)dx + (2x+2y-3)dy = 0 \]
We can rearrange this into the standard form:
\[ \frac{dy}{dx} = -\frac{x+y-1}{2x+2y-3} \]

### Step 2: Simplify the Right-Hand Side
We can factor out a 2 from the denominator:
\[ \frac{dy}{dx} = -\frac{x+y-1}{2(x+y - \frac{3}{2})} \]
Let \( t = x + y \). Then, we can express \( y \) in terms of \( t \):
\[ y = t - x \]
Differentiating with respect to \( x \):
\[ \frac{dy}{dx} = \frac{dt}{dx} - 1 \]

### Step 3: Substitute and Rearrange
Substituting \( y \) and \( \frac{dy}{dx} \) into the equation gives:
\[ \frac{dt}{dx} - 1 = -\frac{(t - 1)}{2(t - \frac{3}{2})} \]
Rearranging this, we have:
\[ \frac{dt}{dx} = -\frac{(t - 1)}{2(t - \frac{3}{2})} + 1 \]

### Step 4: Combine Terms
To combine the terms on the right:
\[ \frac{dt}{dx} = 1 - \frac{(t - 1)}{2(t - \frac{3}{2})} \]
Finding a common denominator:
\[ \frac{dt}{dx} = \frac{2(t - \frac{3}{2}) - (t - 1)}{2(t - \frac{3}{2})} \]
Simplifying the numerator:
\[ 2t - 3 - t + 1 = t - 2 \]
Thus, we have:
\[ \frac{dt}{dx} = \frac{t - 2}{2(t - \frac{3}{2})} \]

### Step 5: Separate Variables
Now we separate the variables:
\[ \frac{2(t - \frac{3}{2})}{t - 2} dt = dx \]

### Step 6: Integrate Both Sides
Integrating both sides:
\[ \int \frac{2(t - \frac{3}{2})}{t - 2} dt = \int dx \]
The left side can be simplified and integrated. After integration, we will have:
\[ 2\log|t - 2| + C_1 = x + C_2 \]

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

### Final Solution
This gives us the implicit solution:
\[ 2\log|x + y - 2| = x + C \]