Solve the differential equation : `dy/(dx)=(x+y+2)/(2(x+y)-1)`

Answer

To solve the differential equation

\[ \frac{dy}{dx} = \frac{x + y + 2}{2(x + y) - 1} \]

we can follow these steps:

### Step 1: Substitution
Let \( u = x + y \). Then, we differentiate both sides with respect to \( x \):

\[ \frac{du}{dx} = \frac{dy}{dx} + 1 \]

This implies:

\[ \frac{dy}{dx} = \frac{du}{dx} - 1 \]

### Step 2: Substitute into the Differential Equation
Substituting \( u \) into the original differential equation gives:

\[ \frac{du}{dx} - 1 = \frac{u + 2}{2u - 1} \]

### Step 3: Rearranging the Equation
Rearranging the equation, we have:

\[ \frac{du}{dx} = \frac{u + 2}{2u - 1} + 1 \]

### Step 4: Simplifying the Right Side
Now, we simplify the right side:

\[ \frac{du}{dx} = \frac{u + 2 + (2u - 1)}{2u - 1} = \frac{3u + 1}{2u - 1} \]

### Step 5: Separating Variables
Now we separate the variables:

\[ (2u - 1) du = (3u + 1) dx \]

### Step 6: Integrating Both Sides
Integrate both sides:

\[ \int (2u - 1) du = \int (3u + 1) dx \]

The left side becomes:

\[ u^2 - u + C_1 \]

And the right side becomes:

\[ 3ux + x + C_2 \]

### Step 7: Combining Constants
Combining the constants, we have:

\[ u^2 - u = 3ux + x + C \]

### Step 8: Substitute Back for \( u \)
Substituting back \( u = x + y \):

\[ (x + y)^2 - (x + y) = 3x(x + y) + x + C \]

### Step 9: Rearranging the Equation
Rearranging gives us a quadratic equation in terms of \( x + y \):

\[ (x + y)^2 - 3x(x + y) - x - C = 0 \]

### Step 10: Final Form
This is a quadratic equation in \( x + y \). You can solve for \( y \) in terms of \( x \) or vice versa depending on the context.