$[\cos(x-y)+\sin(x-y)]dx + [\sin(x-y)-\cos(x-y)]dy$

Question

the differential equation is given, integrating factor is as $\mu(x+y)$. find the integrating factor and solve the equation. $x-y\neq(2k+1)\pi /2$

$[\cos(x-y)+\sin(x-y)]dx + [\sin(x-y)-\cos(x-y)]dy$

can anyone help me? I don't know how to start.

Answer 1

Let $$P=\cos(x-y)+\sin(x-y),\quad Q=\sin(x-y)-\cos(x-y).$$ We find integrating factor $\mu=\mu(x-y)$ from $$\frac{\partial}{\partial y}(P\mu)=\frac{\partial}{\partial x}(Q\mu)$$ We get $$\mu=\frac{1}{\sin(x-y)}$$