Example 1

Here the vector field is {-y, x}. The streamlines should look familiar.

In[1]:=

  p1=PlotVectorField[{-y,x},{x,-5,5},{y,-5,5},Axes->True];

In[2]:=

  DSolve[x'[t]/-y[t]==y'[t]/x[t],y[t],t]

Out[2]=

                             2
  {{y[t] -> -Sqrt[C[1] - x[t] ]}, 
   
                             2
    {y[t] -> Sqrt[C[1] - x[t] ]}}

Look at the solution. We need two branches to make the circle.

To get nice output (instead of a list of solutions, the format is:
Clear the variables.
DSolve[{x'[t]/f1==y'[t]/f2, initial conditions,y,t]
The line y1=Evaluate[y[t]/.Flatten[%][[1]]]/.x[t]->x;
is used to write y as a function x instead of x[t].

In[3]:=

  Clear[x,y,y1,y2];
  DSolve[{x'[t]/-y[t]==y'[t]/x[t],y[0]==0,x[0]==2},y,t];
  y1=Evaluate[y[t]/.Flatten[%][[1]]]/.x[t]->x;
  DSolve[{x'[t]/-y[t]==y'[t]/x[t],y[0]==0,x[0]==2},y,t];
  y2=Evaluate[y[t]/.Flatten[%][[2]]]/.x[t]->x;
  y1
  y2

Out[3]=

             2
  -Sqrt[4 - x ]

Out[4]=

            2
  Sqrt[4 - x ]

In[5]:=

  p2=Plot[{y1,y2},{x,-3,3},AspectRatio->Automatic,
  PlotStyle->{RGBColor[1,0,0]}];

In[6]:=

  Show[p1,p2];

Up to Streamlines