Give the correct answer and solve the given equation (x+y)dx+(x−y)dy=0

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Give the correct answer and solve the given equation   (x+y)dx+(x−y)dy=0

SchulzD · Accepted Answer

First rewrite it as (x−y)dy=−(x+y)dx   (1)Suppose that x−y=0, so y=x. Then0=−2x,so y=x is not a solution of the starding equation. Therefore, y≠x, and we can divide (1) by x−y:dy=−(x+y)x−ydx=x+yy−xdxWe can also write this asy′=dydx=x+yy−x=y−x+2xy−x=1+2xy−x=1+2yx−1          (2)Now use the substitutionu=yx,which we can also write asy−ux,to get thaty′=u′x+u             (3)From(2),y′=1+2yx−1=1+2u−1Combining this and (3) we getu′x+u=1+2u−1Furthermore,u′x=1−u+2u−1=(1−u)(u−1)+2u−1=−(u−1)2+2u−1=−(u−1)2+2u−1Now write u′=dudx:dudx⋅x=−(u−1)2+2u−1Whis is a separable equation,  because we can write it in the form hh(u)du=g(x)dxTo get this, first divide the equation by (u−1)2+2u−1 (notice that this is always nonzero!):u−1(u−1)2+2⋅dudx⋅x=−1Now divide the equation by x:u−1(u−1)2+2⋅dudx=−1xFinally, &quot;multiply&quot; thq equation by dx:

Give the correct answer and solve the given equation displaystyle{left({x}+{y}right)}{left.{d}{x}right.}+{left({x}-{y}right)}{left.{d}{y}right.}={0}

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