Question

Find the particular solution of the differential equation:

`xy(dy)/(dx) = (x + 2) (y + 2)`, given that y(1) = −1.

Answer 1

`xy(dy)/(dx) = (x + 2) (y + 2)`

`y/(y + 2)dy = (x + 2)/xdx`

`int y/(y + 2)dy = int (1 + 2/x)dx`

`int (1 - 2/(y + 2))dy = x + 2 log |x| + C`

y − 2 log |y + 2| = x + 2 log |x| + C

Given y(1) = −1:

−1 − 2 log 1 = 1 + 2 log 1 + C

C = −2

∴ y − 2 log |y + 2| = x + 2 log |x| − 2