Question

Find the general solution of the differential equation:

`log((dy)/(dx)) = ax + by`.

Answer 1

Given differential equation is `log((dy)/(dx)) = ax + by`

⇒ `(dy)/(dx) = e^(ax  +  by)`

⇒ `(dy)/(dx) = e^(ax).e^(by)`

⇒ `(dy)/(e^(by)) = e^(ax) dx`

⇒ `e^(-by) dy = e^(ax) dx`

On integrating both sides, we get

`inte^(-by)dy = inte^(ax)dx`

`e^(-by)/(-b) = e^(ax)/a + C`

⇒ `e^(ax)/a - e^(-by)/b + C` = 0