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 cos2z = cos²z – sin²z = 2cos²z – 1 = 1 – 2sin²z.

⇒

⇒

⇒

We know 1 + cot²x = cosec²x

⇒

⇒

Integrating on both sides we get,

⇒

We know that:

(1) ∫cosec²x = –cotx + C

(2)

(3) ∫adx = ax + C

⇒

⇒

Since z = x – y substituting this we get,

⇒

∴ The solution for the given Differential equation is .