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

Solution:

\((2y - 5)\frac{dy}{dx} = -3\) 

\((2y - 5)dy = -3dx\) 

\(2\cdot\frac{y^2}{2} - 5y = -3x + \lambda\) 

Given: Curve passes through \((0, 1)\)

\(\implies \lambda = -4\) 

Curve Equation: 

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

Vertex of the Parabola: 

\(\left(\frac{3}{4}, \frac{5}{2}\right)\) 

Final Equation:  \(2x + 3y = 9\)