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Question
The solution of the differential equation ydx + (x + xy)dy = 0 is ______.
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Solution
The solution of the differential equation ydx + (x + xy)dy = 0 is xy = ce–y.
Explanation:
The given differential equation is ydx + (x + xy)dy = 0
⇒ (x + xy)dy = – ydx
⇒ x(1 + y)dy = – ydx
⇒ `(1 + y)/y "d"y = - 1/x "d"x`
Integrating both sides, we get
`int (1 + y)/y "d"y = - int 1/x "d"x`
⇒ `int(1/y + 1)"d"y = -int 1/x "d"x`
⇒ log y + y = – log x + log c
⇒ log x + log y + log e y = log c
⇒ log(xy . ey) = log c
∴ xy = ce–y
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