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## Answers

then take y=vx....therefore ,

dy/dx=xdv/dx+v.....now solve step by step....easily the answer will come...

Dividing both the numerator and denominator of the fraction on the right side by x:

y' = [(y/x) - 1]/[(y/x) + 1]

Let y/x = u

==> y = ux ==> y' = u'x + u

The equation becomes

u'x + u = (u - 1)/(u + 1)

u'x = (u - 1)/(u + 1) - u

u'x = -(u² + 1)/(u + 1)

u' = -(u² + 1)/(u + 1)x

[(u + 1)/(u² + 1)]u' = -1/x

[(u + 1)/(u² + 1)]du = -dx/x

udu/(u² + 1) + du/(u² + 1) = -dx/x

2udu/(u² + 1) + 2du/(u² + 1) = -2dx/x

Intergrating each side:

ln(u² + 1) + 2arctan(u) = -ln(x²) + C

ln(u² + 1) + ln(x²) + 2arctan(u) = C

ln(u²x² + x²) + 2arctan(u) = C

ln(y² + x²) + 2arctan(y/x) = C. . . (answer)