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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?

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    $\begingroup$ How about using $u=x+y$ and $x$ as your main variable and solving it by separation? $\endgroup$
    – Maesumi
    Commented May 26, 2013 at 18:08
  • $\begingroup$ Have you learned about how to solve exact ODE's? $\endgroup$
    – Adriano
    Commented May 26, 2013 at 18:09
  • $\begingroup$ does dividing throughout by cos(x+y) help? $\endgroup$
    – dajoker
    Commented May 26, 2013 at 18:10

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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 $y=-x+\arccos(K/x)$.

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  • $\begingroup$ @Alejandro Continuing from here, the equation becomes $\frac{dx}{du}-x\tan{u}=0$, which is a first order linear ODE. $\endgroup$ Commented May 26, 2013 at 19:00

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