Solve the following differential equations:

(x + y)(dx–dy) = dx + dy

Given Differential equation is:

⇒ (x + y)(dx–dy) = dx + dy

⇒ (x + y)dx –(x + y)dy = dx + dy

⇒ (x + y–1)dx = (x + y + 1)dy

⇒ ……(1)

Let us assume z = x + y

Differentiating w.r.t x on both sides we get,

⇒

⇒

⇒ ……(2)

Substituting (2) in (1) we get,

⇒

⇒

⇒

⇒

Bringing like variables on same side(i.e., variable seperable technique) we get,

⇒

⇒

⇒

⇒

Integrating on both sides we get,

⇒

We know that:

(1) ∫ adx = ax + C

(2)

⇒

Since z = x + y we substitute this,

⇒ x + y + log(x + y) = 2x + C

⇒ y + log(x + y) = x + C

∴ The solution for the given Differential equation is y + log(x + y) = x + C.