Flow Lines of Vector Field F = sec(x) i + k

[stripes]

Homework Statement

Define the vector field F = sec(x) i + k
(a) Express the flow lines of F in equations form.

(b) Express in equations form the particular flow line through the point (0, 3, 2).

My next question is a bonus question. I'm just reading up on this now but if someone could kind of give me an intuitive grasp on the ideas related to the question, that'd be great. Mainly my question is the one above, not so much the bonus (though I could really use the marks).

Homework Equations

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The Attempt at a Solution

I have found that the flow lines are z = sin(x) + C, for some constant C. I imagine this is in "equations form" (as opposed to parametric form). Now since I missed the class where we discussed flow lines of a vector field, I am stuck on the next question.

I have tried looking up some resources online, and I have seen some websites say I need to solve a system of ordinary differential equations. But I'm unsure how to arrive at such a situation, and we don't discuss ODEs in this course.

We have no functions of y; the vector field only has a function of x, and the equation of the flow lines also has no y in it. Where do I start?

 

[haruspex]

Not sure whether 'equations form' is as opposed to parametric or as opposed to vector. Anyway, it's clearly not vector form, so to define a line in 3-space will require two equations.
Based on what you have worked out, how will y interact with x and z in these equations?

 

[MednataMiza]

I will help you with a couple o'plots:

First: Plot of all flow lines / you got that right /

Second: Plot of all flow lines + plot of the equation which runs through the point (0,3,2)