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Question
For the differential equation `xy(dy)/(dx) = (x + 2)(y + 2)` find the solution curve passing through the point (1, –1).
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Solution
Given, xy `dy/dx` = (x + 2)(y + 2)
`=> y/((y + 2)) dy = (x + 2)/x dx`
`=> (1 - 2/(y + 2)) dy = (1 + 2/x) dx`
On integrating
`int (1 - 2/(y + 2)) dy = int (1 + 2/x) dx`
y - 2 log (y + 2) = x + 2 log x + C …(i)
∵ The curve passes through the point (1, -1) so x = 1, y = -1
∴ -1 - 2 log (1) = 1 + 2 log (1) + C [∵ log 1 = 0]
-1 = 1 + C
⇒ C = -2
On putting C = – 2 in equation (i)
y - 2 log (y + 2) = x + 2 log x + 2
⇒ y - x + 2 = 2 log x + 2 log (y + 2)
⇒ y - x’ + 2 = 2 [log x (y + 2)]
y - x + 2 = log [x2 (y + 2)2]
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