How Do Field Lines Relate to Differential Equations in Vector Calculus?

[mewmew]

I am going through the first chapter of div,grad,curl and all that and am a bit confused about problem 6. The question is:

Show that the field lines y=y(x) of a vector function

$$F(x,y)=F_x(x,y)i+F_y(x,y)j$$ are solutions of the differential equation $$\frac{dy} {dx} = \frac{F_y(x,y)} {F_x(x,y)}$$

I am not very good with calculus do to some conflict in my schedule that has forced me to put off multivariable calculus(I do know the basics of it though and can do simple 2 variable differentiation). Can anyone give me any guidance? I am not even really sure I understand the problem very well so I am not sure how to start, the y=y(x) is sort of throwing me off. Thanks

 

[robphy]

dy/dx is the derivative (i.e. slope-function) of the curve y(x).

At a given point (x,y), what does the ratio of these vector components F_y/F_x represent?