the theory says that even without having the solution of the diff equation

for any given (x,y) we could know the slope of the solution at this point.

if we draw a line threw every point with the solpe of the solution we get the photo

if we find a line wich is tangent to the lines then its the solution

if we found the solution then in each one of its points its tangent to the collection of the grath.

for this equation

y'=y/x

the direction graph is given by the following photo.

i dont know how they made this graph.

i dont have the idea of how to find the slopes at point (x,y)

we have y/x x cannot be 0 thats it

that line collection is some collection of lines y=x y=2x y=-x etc...

i dont know how it correspond to y/x