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

JEE Main · 2024 · 09 Apr 2024 · Shift 1 · Differential Equations

The solution curve, of the differential equation

2 y \frac{dy}{dx} + 3 = 5 \frac{dy}{dx}

, passing through the point

(0, 1)

is a conic, whose vertex lies on the line :

Answer: (A)

2 y \frac{d y}{d x} + 3 = 5 \frac{d y}{d x} \Rightarrow (2 y - 5) d y = - 3 d x \Rightarrow y^{2} - 5 y = - 3 x + C

. Through

(0, 1)

1 - 5 = C \Rightarrow C = - 4

, so

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

. The vertex is

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

, and

2 x + 3 y = \frac{3}{2} + \frac{15}{2} = 9

— option (1).

JEE Main · 2024 · 09 Apr 2024 · Shift 1 · Single correct option

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