The equation of the curve passing through the point (1,1) and satisfying the differential equation `(dy)/(dx) = (x+2y-3)/(y-2x+1)` is

Options

A. `x^2-4xy-y^2+6x+2y-4=0`
B. `x^2+4xy-y^2-6x+2y+4=0`
C. `x^2+4xy-y^2-6x-2y+4=0`
D. `x^2+4xy+y^2-6x-2y-4=0`

Answer

To solve the problem, we need to find the equation of the curve that passes through the point (1, 1) and satisfies the given differential equation:

\[ \frac{dy}{dx} = \frac{x + 2y - 3}{y - 2x + 1} \]

### Step 1: Cross Multiply
We start by cross-multiplying the differential equation:

\[ (y - 2x + 1) dy = (x + 2y - 3) dx \]

### Step 2: Rearranging Terms
Next, we rearrange the equation to separate the variables:

\[ y \, dy - 2x \, dy + dy = x \, dx + 2y \, dx - 3 \, dx \]

This simplifies to:

\[ (y + 1) dy - (x - 3) dx = 2y \, dx - 2x \, dy \]

### Step 3: Grouping Terms
Now we group the terms involving \(y\) on one side and those involving \(x\) on the other side:

\[ (y + 1) dy = (x - 3) dx + 2y \, dx - 2x \, dy \]

### Step 4: Integrate Both Sides
Next, we integrate both sides. The left side integrates to:

\[ \int (y + 1) dy = \frac{y^2}{2} + y + C_1 \]

The right side integrates to:

\[ \int (x^2/2 - 3x + 2xy) dx = \frac{x^2}{2} - 3x + xy + C_2 \]

### Step 5: Combine Constants
Combining the constants from both sides, we get:

\[ \frac{y^2}{2} + y = \frac{x^2}{2} - 3x + xy + C \]

### Step 6: Rearranging the Equation
Rearranging gives us:

\[ y^2 + 2y - 2xy + 2x^2 - 6x + 2C = 0 \]

### Step 7: Substitute the Point (1, 1)
Now we substitute the point (1, 1) into the equation to find the constant \(C\):

\[ 1^2 + 2(1) - 2(1)(1) + 2(1)^2 - 6(1) + 2C = 0 \]

This simplifies to:

\[ 1 + 2 - 2 + 2 - 6 + 2C = 0 \implies -3 + 2C = 0 \implies 2C = 3 \implies C = \frac{3}{2} \]

### Step 8: Final Equation
Substituting \(C\) back into the equation gives:

\[ y^2 + 2y - 2xy + 2x^2 - 6x + 3 = 0 \]

This is the equation of the curve that passes through the point (1, 1) and satisfies the given differential equation.

### Final Answer
The equation of the curve is:

\[ y^2 - 2xy + 2x^2 + 2y - 6x + 3 = 0 \]