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Question

Question: If y = ƒ(x) is the equation of the curve and its differential equation is given by \(\frac{dy}{dx}\)...

If y = ƒ(x) is the equation of the curve and its differential equation is given by dydx\frac{dy}{dx}= x+2y+3\frac{x + 2}{y + 3}, then the equation of the curve, if it passes through (2, 2), is –

A

x2 – y2 + 4x – 6y + 4 = 0

B

x2 – y2 + 4x + 6y = 0

C

x2 – y2 – 4x – 6y = 0

D

x2 – y2 – 4x – 6y – 4 = 0

Answer

x2 – y2 + 4x – 6y + 4 = 0

Explanation

Solution

The given equation is (y + 3) dy = (x + 2) dx

Ž y22\frac{y^{2}}{2} + 3y = x22\frac{x^{2}}{2} + 2x + c

Since, it passes through (2, 2)

\ 2 + 6 = 2 + 4 + c Ž c = 2 \ y22\frac{y^{2}}{2} + 3y = x22\frac{x^{2}}{2} + 2x + 2 Ž y2 + 6y = x2 + 4x + 4

Ž x2 + 4x – y2 – 6y + 4 = 0