The solution curve, of the differential equation `2y (dy)/(dx) + 3 = 5(dy)/(dx)`, passing through the point `(0, 1)` is a conic, whose vertex lies on the line:

Options

A. `2x + 3y = 9`
B. `2x + 3y = -6`
C. `2x + 3y = 6`
D. `2x + 3y = -9`

Answer

To solve the differential equation \( 2y \frac{dy}{dx} + 3 = 5 \frac{dy}{dx} \) and find the conic whose vertex lies on a specific line, we can follow these steps:

### Step 1: Rearranging the Differential Equation
We start with the given differential equation:
\[ 2y \frac{dy}{dx} + 3 = 5 \frac{dy}{dx} \]
Rearranging this gives:
\[ 2y \frac{dy}{dx} - 5 \frac{dy}{dx} + 3 = 0 \]
Factoring out \( \frac{dy}{dx} \):
\[ \left(2y - 5\right) \frac{dy}{dx} + 3 = 0 \]
This can be rewritten as:
\[ \frac{dy}{dx} = -\frac{3}{2y - 5} \]

### Step 2: Separating Variables
Now we separate the variables:
\[ (2y - 5) dy = -3 dx \]

### Step 3: Integrating Both Sides
Next, we integrate both sides:
\[ \int (2y - 5) dy = \int -3 dx \]
The left side integrates to:
\[ y^2 - 5y + C_1 \]
The right side integrates to:
\[ -3x + C_2 \]
Combining these results gives:
\[ y^2 - 5y = -3x + C \]

### Step 4: Finding the Constant of Integration
We know that the curve passes through the point \( (0, 1) \). Substituting \( x = 0 \) and \( y = 1 \):
\[ 1^2 - 5(1) = -3(0) + C \]
This simplifies to:
\[ 1 - 5 = C \implies C = 4 \]
Thus, our equation becomes:
\[ y^2 - 5y = -3x + 4 \]

### Step 5: Rearranging the Equation
Rearranging gives us:
\[ y^2 - 5y + 3x - 4 = 0 \]

### Step 6: Completing the Square
To express this as a conic, we complete the square for the \( y \) terms:
\[ y^2 - 5y = (y - \frac{5}{2})^2 - \frac{25}{4} \]
Substituting this back into the equation:
\[ (y - \frac{5}{2})^2 - \frac{25}{4} + 3x - 4 = 0 \]
This simplifies to:
\[ (y - \frac{5}{2})^2 = -3x + \frac{41}{4} \]

### Step 7: Identifying the Vertex
The vertex of this conic is at:
\[ \left(\frac{41}{12}, \frac{5}{2}\right) \]

### Step 8: Finding the Line Equation
To find the line on which this vertex lies, we can check the equation \( 2x + 3y = k \) for various values of \( k \):
Substituting the vertex coordinates:
\[ 2\left(\frac{41}{12}\right) + 3\left(\frac{5}{2}\right) = \frac{82}{12} + \frac{15}{2} = \frac{82 + 90}{12} = \frac{172}{12} = \frac{43}{3} \]
Thus, the line equation is:
\[ 2x + 3y = \frac{43}{3} \]

### Final Answer
The vertex of the conic lies on the line \( 2x + 3y = \frac{43}{3} \).

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