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.