Question

For the differential equation `xy(dy)/(dx) = (x + 2)(y + 2)`  find the solution curve passing through the point (1, –1).

Answer 1

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]