Question

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

• A. \( 2x + 3y = 9 \)
• B. \( 2x + 3y = -9 \)
• C. \( 2x + 3y = -6 \)
• D. \( 2x + 3y = 6 \)

Answer 1

The Correct Option is A

To solve the given differential equation and determine the line containing the vertex of the resultant conic, we first simplify and solve the differential equation.

The provided differential equation is: \(2y \frac{dy}{dx} + 3 = 5 \frac{dy}{dx}\)

Rearrange terms to isolate the derivative:

\(2y \frac{dy}{dx} - 5 \frac{dy}{dx} = -3\)

Factor out \(\frac{dy}{dx}\):

\(\left( 2y - 5 \right) \frac{dy}{dx} = -3\)

Solve for \(\frac{dy}{dx}\):

\(\frac{dy}{dx} = \frac{-3}{2y - 5}\)

This is a separable differential equation. Separate the variables:

\((2y - 5) \, dy = -3 \, dx\)

Integrate both sides:

\(\int (2y - 5) \, dy = -\int 3 \, dx\)

The integration yields:

\(y^2 - 5y = -3x + C\)

Rearrange to form the equation of the conic:

\(y^2 - 5y + 3x = C\)

Utilize the initial condition that the curve passes through \((0, 1)\) to find the constant C.

Substitute \((x, y) = (0, 1)\) into the equation:

\(1^2 - 5 \times 1 + 3 \times 0 = C\)

Calculate the value of C:

\(1 - 5 = C \Rightarrow C = -4\)

The equation of the conic is therefore:

\(y^2 - 5y + 3x = -4\)

To find the vertex of this conic, complete the square for the terms involving \(y\):

\(y^2 - 5y = -(3x + 4)\)

Complete the square for the \(y\) terms:

\(y^2 - 5y + \left(\frac{5}{2}\right)^2 = \left(\frac{5}{2}\right)^2 - (3x + 4)\)

Simplify the equation:

\(\left(y - \frac{5}{2}\right)^2 = \frac{25}{4} - 3x - 4\)

Further simplification gives:

\(\left(y - \frac{5}{2}\right)^2 = \frac{9}{4} - 3x\)

The vertex of this conic section is at \((x, y) = (h, \frac{5}{2})\). At the vertex, the left side of the equation must be zero.

\(\frac{9}{4} = 3h\)

Solve for \(h\):

\(h = \frac{3}{4}\)

Thus, the vertex lies on the horizontal line \(y = \frac{5}{2}\).

Verify the vertex lies on the line \(2x + 3y = 9\) by plugging in the vertex coordinates \(\left(\frac{3}{4}, \frac{5}{2}\right)\).

Substitute \(x = \frac{3}{4}\) and \(y = \frac{5}{2}\) into the line equation:

\(2 \times \frac{3}{4} + 3 \times \frac{5}{2} = 9\)

Perform the calculation:

\(\frac{3}{2} + \frac{15}{2} = 9\)

\(9 = 9\)

The verification confirms that the correct line on which the vertex lies is \(2x + 3y = 9\).