Question

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

Answer 1

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`