Thread: Slope field primer

Alright. I don't have a picture of it. What I'm given is a slope field from -1<x<1 and -1<y<1 and told to graph it for A. y(-0.5)=1 and B. y(0)=1.

What I don't understand is what the y(-0.5)=1 and y(0)=1 are asking. Is my line supposed to cross at x=-.5 and Y=1? What do I do with the rest of it then if I'm only given that single point?

Basically, how do I read a slope field and how do I start drawing in it?

If your differential equation y' = f(x,y) satisfies certain regularity conditions, then you're guaranteed a unique solution, which means, among other things, that no two integral curves will intersect. With slope fields, if you're asked to graph a solution of the DE on it, you start at the initial condition, say y(-0.5) = 1, and draw a curve such that the slopes in the slope field are always tangent to your curve. See here for an example. You're given two different initial conditions; if your slope field doesn't have any "intersections" in it, then you should get two separate integral curves (solution curves) for the corresponding DE. Does that make sense?

Originally Posted by Wolvenmoon

Alright. I don't have a picture of it. What I'm given is a slope field from -1<x<1 and -1<y<1 and told to graph it for A. y(-0.5)=1 and B. y(0)=1.

What I don't understand is what the y(-0.5)=1 and y(0)=1 are asking. Is my line supposed to cross at x=-.5 and Y=1? What do I do with the rest of it then if I'm only given that single point?

Basically, how do I read a slope field and how do I start drawing in it?

Here is a slope field




One way to visualize this is to think of the slope field as the velocity of a river. Now if you drop a cork on the point given where will the current take it.

Okay, seeing the arrow points helped immensely. I just have | and \ and /s on mine.

So to cement it in my head, he's giving me points to start on. My graph always stays parallel to the lines in the slope field, and that's about all there is to it?

Originally Posted by Wolvenmoon

Okay, seeing the arrow points helped immensely. I just have | and \ and /s on mine.

So to cement it in my head, he's giving me points to start on. My graph always stays parallel to the lines in the slope field, and that's about all there is to it?

Pretty much. There's a lot less to slope fields than meets the eye.

TES: For me your picture is broken. Any chance you could fix?

Originally Posted by Ackbeet

Pretty much. There's a lot less to slope fields than meets the eye.

TES: For me your picture is broken. Any chance you could fix?

Hmm it is working for me I will upload another copy here

TES: Nice!