Solving the next differential equation

How can I solve the next differential equation? $$\cos(x+y)dx=x\sin(x+y)dx+x\sin(x+y)dy$$ I dont know what kind of equation it is. It's not homogeneous, separable differential equation or linear. Any suggestions?

• How about using $u=x+y$ and $x$ as your main variable and solving it by separation? – Maesumi May 26 '13 at 18:08
• Have you learned about how to solve exact ODE's? – Adriano May 26 '13 at 18:09
• does dividing throughout by cos(x+y) help? – dajoker May 26 '13 at 18:10

use $u=x+y$, $y=u-x, dy =du-dx$ to get $\cos u dx =x \sin u dx +x \sin u (du-dx)$ which simplifies to $\cos u dx=x\sin u du$.
So $dx/x= \sin u du /\cos u$, or $\ln x= -\ln \cos u +C$, or $x= K/\cos u$, or $x\cos(x+y)=K$, or $x+y=\arccos (K/x)$, or $y=-x+\arccos(K/x)$.
• @Alejandro Continuing from here, the equation becomes $\frac{dx}{du}-x\tan{u}=0$, which is a first order linear ODE. – Ataraxia May 26 '13 at 19:00