Solve the following differential equations:

Given Differential equation is:

⇒ ……(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)

⇒ 2z – log(z + 3) = x + C

Since z = x – y, we substitute this,

⇒ 2(x – y) – log(x–y + 3) = x + C

⇒ 2x – 2y –log(x–y + 3) = x + C

⇒ x – 2y –log(x–y + 3) = C

∴ The solution for the given Differential equation is: x – 2y –log(x–y + 3) = C.