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

Answer: (A)

$2x + 3y = 9$

Answer 2

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: **$2x + 3y = 9$