Slope of the tangent to the curve at P(x,y) is given by `(3y + 2x + 4)/(4x + 6y+ 5)`. If the curve passes through `(0, -1)`, find its equation

Answer

To find the equation of the curve given the slope of the tangent, we can follow these steps:

### Step 1: Set up the differential equation
The slope of the tangent to the curve at point \( P(x, y) \) is given by:
\[ \frac{dy}{dx} = \frac{3y + 2x + 4}{4x + 6y + 5} \]

### Step 2: Separate variables
We can rewrite the equation to separate the variables \( y \) and \( x \):
\[ (4x + 6y + 5) dy = (3y + 2x + 4) dx \]

### Step 3: Rearrange the equation
Rearranging gives us:
\[ (4x + 6y + 5) dy - (3y + 2x + 4) dx = 0 \]

### Step 4: Integrate both sides
Now we need to integrate both sides. However, to simplify the integration, we can express it in a more manageable form. We can use substitution if necessary, but for now, we will integrate directly:
\[ \int (4x + 6y + 5) dy = \int (3y + 2x + 4) dx \]

### Step 5: Solve the integrals
Integrating both sides:
- The left side:
\[ \int (4x + 6y + 5) dy = 4xy + 3y^2 + 5y + C_1 \]
- The right side:
\[ \int (3y + 2x + 4) dx = 2xy + 4x + C_2 \]

### Step 6: Combine and simplify
Setting the two integrals equal to each other gives us:
\[ 4xy + 3y^2 + 5y = 2xy + 4x + C \]
where \( C = C_2 - C_1 \).

### Step 7: Rearranging the equation
Rearranging the equation, we have:
\[ 4xy - 2xy + 3y^2 + 5y - 4x - C = 0 \]
This simplifies to:
\[ 2xy + 3y^2 + 5y - 4x - C = 0 \]

### Step 8: Substitute the point (0, -1)
Since the curve passes through the point \( (0, -1) \), we can substitute \( x = 0 \) and \( y = -1 \) into the equation:
\[ 2(0)(-1) + 3(-1)^2 + 5(-1) - 4(0) - C = 0 \]
This simplifies to:
\[ 0 + 3 - 5 - C = 0 \implies -2 - C = 0 \implies C = -2 \]

### Step 9: Final equation of the curve
Substituting \( C \) back into the equation gives:
\[ 2xy + 3y^2 + 5y - 4x + 2 = 0 \]

### Final Answer
The equation of the curve is:
\[ 2xy + 3y^2 + 5y - 4x + 2 = 0 \]