Solve the following differential equations:

Given Differential Equation is :

⇒

⇒ ……(1)

Let us assume z = x + y + 1

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)

⇒ z – log(z + 1) = x + C

Since z = x + y we substitute this,

⇒ x + y–log(x + y + 1) = x + C

⇒ y–log(x + y + 1) = C

⇒ y = log(x + y + 1) + C

∴ The solution for the given Differential Equation is y = log(x + y + 1) + C.