Show that the differential equation `xcos(y/x)(dy)/(dx)=ycos(y/x)+x` is homogeneous and solve it.

Answer

`sin (y/x)=log |x|+c`

Explanation

We are given

`xcos(y/x)(dy)/(dx)=ycos(y/x)+x`


Simplifying, we get


`(dy)/(dx)=(ycos(y/x)+x)/(xcos(y/x))-(1)`


now the equation is homogenous as

if `F(x,y)=(ycos(y/x)+x)/(xcos(y/x))` then

`F(Ax,Ay)=(Aycos(Ay/Ax)+Ax)/(Axcos(Ay/Ax))`

`=(Aycos(y/x)+Ax)/(Axcos(y/x))`

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