The Solution curve of the differential equation 2 y = d y d x + 3 = 5 d y d x , passing through the point (0, 1) is a conic, whose vertex lies on the line: [2024]

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Solution

Q.
The Solution curve of the differential equation $2 y = \frac{d y}{d x} + 3 = 5 \frac{d y}{d x}$ , passing through the point (0, 1) is a conic, whose vertex lies on the line: [2024]

1 2x + 3y = –6

2 2x + 3y = 6

3 2x + 3y = –9

4 2x + 3y = 9

Ans.
(4)

We have, $2 y \frac{d y}{d x} + 3 = 5 \frac{d y}{d x}$

$\Rightarrow$ 2ydy + 3dx = 5dy

Integrating both sides, we get

$y^{2} + 3 x = 5 y + c$

Now, at point (0, 1) i.e., at x = 0, y = 1, we have

1 + 0 = 5 + c $\Rightarrow$ c = –4

So, $y^{2} - 5 y = - 3 x - 4$

$\Rightarrow y^{2} - 5 y + \frac{25}{4} - \frac{25}{4} = - 3 x - 4$

$\Rightarrow {(y - \frac{5}{2})}^{2} = - 3 x + \frac{9}{4} \Rightarrow {(y - \frac{5}{2})}^{2} = - 3 (x - \frac{3}{4})$

Vertex = $(\frac{3}{4}, \frac{5}{2})$ , Which satisfies by 2x + 3y = 9.

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