Q:

How to correctly use Dsolve?

how can I Plot these Equations?
1)
(x-y[x])y'[x]== x+y[x]

2)
Sol = DSolve[{1/r^2*D[r^2*ψ[r], r] + Subscript[ν, ct] ψ[r] == 0, ψ[r0] == ψ0/(r0^2 Ω)}, 
    ψ[r], r] /. r0 -> 0 /. Subscript[ν, ct] -> 1/λct;

A:

 sol=DSolve[(x-y[x])y'[x]== x+y[x],y[x],x]

 ContourPlot [ Evaluate@(First@sol/.C[1]->0/.y[x]->y) ,{x,0,2}, {y,-1,5}]

ContourPlot [ Evaluate@Table[First@sol/.C[1]->i/.y[x]->y,{i,-2,2 }] ,{x,0,2}, {y,-1,5}]

incidentally, this throws a spurious unable to solve warning.. (yet shows the plot just fine anyway ).

Edit: here is how to do it cleanly..
sol = DSolve[(x - y[x]) y'[x] == x + y[x], y[x], x]
sol = (List @@ sol)[[1]] /.  y[x] -> y  /. C[1] -> i
ContourPlot[Evaluate[Table[sol, {i, -2, 2}]], {x, 0, 2}, {y, -1, 5}]

