Differential Equations: Direction Field

In summary, your book will allow you to input values for x and y and compute a whole bunch of slopes, but you should do this by hand first and only look at a computer generated plot after you understand what's happening.
  • #1
ineedhelpnow
651
0
Sketch a direction field for the differential equation. Then use it to sketch three solution curves.
$y'=x-y+1$

I really need help drawing this, I'm super confused. :confused:
 
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  • #2
ineedhelpnow said:
Sketch a direction field for the differential equation. Then use it to sketch three solution curves.
$y'=x-y+1$

I really need help drawing this, I'm super confused. :confused:

The so called 'direction field' is simply the mapping of the sign of y' in the x-y plane. So setting y' = m = const, You obtain...

y' = m -> y = x + 1 - m (1)

The (1) means that in all the points where y > x + 1 is y' < 0, in all the points where y = x + 1 is y' = 0 and in all the points where y < x + 1 is y' > 0...

Kind regards

$\chi$ $\sigma$
 
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  • #3
what my book does is plug in values for x and y into the equation and then they compute a whole bunch of different slopes. Should I make my graph from like $-2\le x \le 2$ and $-2\le y \le 2$ ?
 
  • #4
is there a way to do it on my calculator?
 
  • #5
ineedhelpnow said:
what my book does is plug in values for x and y into the equation and then they compute a whole bunch of different slopes. Should I make my graph from like $-2\le x \le 2$ and $-2\le y \le 2$ ?

That's what I would do...and for simplicity only compute the slope at lattice points, that is, those points whose coordinates are integers. This will give you 25 points at which to compute a slope.

I imagine your calculator can draw direction fields...back when I was a student, we had to program our calculators to do this. :D
 
  • #6
oh i think i got it
 
  • #7
i had it but i mistakenly closed the document and i keep putting in the equation but it won't show up anymore. I am putting in the equation of the graph but it's also asking me for initial conditions.
 
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  • #8
I really recommend you do this problem by hand first, and only then look at a computer generated plot of the field. You get much more of a feel for what's going on by actually getting in there and doing it yourself. :D
 

FAQ: Differential Equations: Direction Field

What is a direction field for a differential equation?

A direction field is a graphical representation of the solutions to a differential equation. It shows the direction in which the solution curves will move at various points in the graph.

How is a direction field created?

A direction field is created by plotting a series of small line segments at different points on the graph, with the slope of each line segment corresponding to the slope of the solution curve at that point.

What is the purpose of a direction field?

The purpose of a direction field is to give a visual understanding of the behavior of a differential equation. It can help in predicting the general shape and direction of solution curves.

How can a direction field be used to solve a differential equation?

A direction field can be used to estimate the behavior of the solution to a differential equation at any given point. By following the direction of the line segments, one can approximate the solution curve at that point.

Can a direction field accurately represent all solutions to a differential equation?

No, a direction field can only give a general understanding of the behavior of the solutions. It cannot accurately represent all possible solutions, as some may be complex or behave in unexpected ways.

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