The general solution of the differential equation d(1+yx)+dydx = 0 is ______. - Mathematics

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The general solution of the differential equation `(1 + y/x) + ("d"y)/(d"x)` = 0 is ______.

Options

  • x2 + y2 = c

  • 2x2 – y2 = c

  • x2 + 2xy = c

  • y2 + 2xy = c

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Solution

The general solution of the differential equation `(1 + y/x) + ("d"y)/(d"x)` = 0 is x2 + 2xy = c.

Explanation:

`(1 + y/x) + ("d"y)/(d"x)` = 0 

⇒ `("d"y)/("d"x)= - (1 + y/x)`  ......(i)

Put v = `y/x`

⇒ y = x   ......(ii)

⇒ `("d"y)/("d"x) = "v" + x "dv"/("d"x)`   ......(iii)

Substituting (ii) and (iii) in (i), we get

`"v" + x "dv"/("d"x)` = – 1 – v

⇒ `x "dv"/("d"x)` = – 1 – 2v

Integrating on both sides, we get

`int "dv"/(1 + 2"v") = - int ("d"x)/x + log"c"_1`

⇒ `1/2 log (1 + 2"v") =  logx + log"c"_1`

⇒ `log(1 + 2 y/x) = 2 log  "c"_1/x`

⇒ `(x + 2y)/x = ("c"_1/x)^2`

⇒ `x^2 + 2xy = "c"_1^2`

⇒ x2 + 2xy = c, where c = `"c"_1^2`

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