Question

The general solution of e^x cosy dx – e^x siny dy = 0 is ______.

Options

• e^x cosy = k

• e^x siny = k

• e^x = k cosy

• e^x = k siny

Answer 1

The general solution of e^x cosy dx – e^x siny dy = 0 is e^x cosy = k.

Explanation:

The given differential equation is e^x cosy dx – e^x siny dy = 0

⇒ e^x (cosy dx – siny dy) = 0

⇒ cosy dx – siny dy = 0   ......[∵ e^x ≠ 0]

⇒ siny dy = cosy dx

⇒ `siny/cosy "d"y` = dx

Integrating both sides, we have

`int siny/cosy "d"y = int "d"x`

⇒ `-log|cosy| = x + log "k"`

⇒ `log  1/cosy - log "k"` = x

⇒ `log(1/("k" cosy))` = x

⇒ `1/("k" cosy)` = e^x

⇒ `1/"k"` = e^x cosy

⇒  e^x cosy = c  .....`["c" = 1/"k"]`