Solve the differential equation
`x dy-y dx=sqrt(x^(2)+y^(2)) dx`.

Explanation

`" "x (dy)/(dx) -y= sqrt( x^(2) +y^(2))`
`" " rArr x (dy)/(dx) = y + sqrt(x^(2)+y^(2))`
`rArr (dy)/(dx) = (y+ sqrt(x^(2) + y^(2)))/(x)...................(1) `
`" " ` let y = vx
differentiating with w.r.t x
`" " rArr (dy)/(dx) = v + x(dv)/(dx) `
put in (1)
`rArr v + x (dv)/(dx) = vx+ sqrt(x^((2) + v^(2)x^(2))) /(x)`
`rArr v + x (dv)/(dx) = (x(v+ sqrt(1+v^(2))))/(x)`
`rArr x (dv)/(dx) = v + sqrt(1+ v^(2))-v`
`rArr x (dv)/(dx) = sqrt(1+v^(2))`
`rArr (dv)/(sqrt(1+v^(2))) = (dx)/(x)`
integrating both sides
`rArr int (dv)/(sqrt(1+v^(2))) = int (dx)/(x)`
`rArr log(v+ sqrt(1+v^(2))) = log x+ logc`
`rArr log(v + sqrt(1+ v^(2))) = log cx`
`" " rArr (v + sqrt(1+ v^(2)))= cx`
` rArr ((y)/(x) + sqrt(1 + ((y)/(x))^(2)))= cx`
`" " rArr y + sqrt(x^(2) + y^(2)) = cx^(2)`