Solve the differential equation `( x + y) dy + (x - y) dx = 0`; given that y = 1 when x = 1.

Explanation

`x+ydy=(y-x)dx`

`frac{dy}{dx}=frac{y-x}{y+x}`

Let `y=vx`

`frac{dy}{dx}=v+x frac{dv}{dx}`

`v+xfrac{dv}{dx}=frac{vx-x}{vx+x}`

`x frac{dv}{dx}=frac{v-1}{v+1}-v`

`int frac{1+v}{1+v^2}dv=int -dx/x`

`tan^(-1)=1/2 log abs(1+v^2)=-logx+logC`
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