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,

⇒

We know that

⇒

⇒

⇒

⇒

⇒

⇒

⇒

Integrating on both sides we get,

⇒

We know that:

(1)

(2) ∫adx = ax + C

⇒ z + log(cosz + sinz) = 2x + C

Since z = x + y, we substitute this,

⇒ x + y + log(cos(x + y) + sin(x + y)) = 2x + C

⇒ y + log(cos(x + y) + sin(x + y)) = x + C

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