Graphical represtation of a diff equation solution

[nhrock3]

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 which 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 don't know how they made this graph.
i don't have the idea of how to find the slopes at point (x,y)

we have y/x x cannot be 0 that's it

that line collection is some collection of lines y=x y=2x y=-x etc...
i don't know how it correspond to y/x

 

[Mark44]

On the line y = x, y/x = 1, except when x = 0.
On the line y = 2x, y/x = 2, except when x = 0.
On the line y = kx, y/x = k, except when x = 0.
It's pretty straightforward.

 

[nhrock3]

thnks :)