I fed this equation into the Matlab terminal:
eqn = y/s-i/s == (y^2*w^2/v^2-w^2)^(1/2)
eqn =

sol = solve(eqn, y)
Warning: Solutions are only valid under certain conditions. To include parameters and conditions in the solution, specify the 'ReturnConditions' value as 'true'.
sol =

sol1 = solve(eqn, y, 'returnconditions', true)
sol1 =
y: [2x1 sym]
parameters: [1x0 sym]
conditions: [2x1 sym]
sol1.y
ans =

and got these two solutions, the two intersection points of a hyperbola with a straight line:
-(v*(i*v + s*w*(i^2 + s^2*w^2 - v^2)^(1/2)))/(s^2*w^2 - v^2)
-(v*(i*v - s*w*(i^2 + s^2*w^2 - v^2)^(1/2)))/(s^2*w^2 - v^2)
the first solution did the trick but the second only after I removed the minus sign at it's beginning - that's besides the next minus sign that replaces the plus in the first solution
a big thank you for all the support I received from you in the past