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Solution :

` (dy)/(dx) = y + sqrt(x^(2) +y^(2))/x ` <br> Put y =vx ` therefore (dy)/(dx) =v +x (dv)/(dx)` <br> (1) becomes ` v + x(dv)/(dx) = vx +sqrt(x^(2)+v^(2)x^(2))/x` <br> ` v+ x (dv)/(dx) = v + sqrt(1 +v^(2))` <br> ` x(dv)/(dx) =sqrt(1+v^(2)) " " therefore 1/(sqrt(1+v^(2)) dv = 1/x dx` <br> Integrating we get, <br> ` int 1/(sqrt(1 +v^(2)) dv = int 1/x dx +c_(1)` <br> ` therefore log|v+ sqrt(1+v^(2)) | = log |x| + log c, " where " c_(1) = log c` <br> ` log |y/x + sqrt(1+y^(2)/x^(2)| = log |cx|` <br> ` (y + sqrt(x^(2)+y^(2)))/x =cx " " therefore y + sqrt(x^(2) +Y^(2)) =cx^(2) ` <br> This is the general solution.