I Determining directional Field for say ##\dfrac{dy}{dx}=y-x##

[chwala]

TL;DR Summary

What are the key factors to consider when determining the directional fields for curves/straight lines?

I am looking at this now : My understanding is that in determining the directional fields for curves; establishing the turning points and/or inflection points if any is key...then one has to make use of limits and check behaviour of function as it approaches or moves away from these points thus giving us a family of curves. For the question posted here , ...we set $\dfrac{dy}{dx}=0, c=y-x$ and then proceed to get a family of curves (isoclines) then proceed to check the behaviour of other values in the neighbourhood of the parallel lines.

Pretty clear but i may have missed on something hence my post.

 

[fresh_42]

I'm not sure I understand what you want to know. However, I have a few interesting links:

• Theory
  https://en.wikipedia.org/wiki/Euler_method
  https://en.wikipedia.org/wiki/Heun's_method
  https://en.wikipedia.org/wiki/Runge–Kutta_methods
• Solution
  https://www.wolframalpha.com/input?i=y'=y-x
• Playground
  http://elsenaju.info/Rechner/DGL-1-Ordnung.htm
  https://planetcalc.com/8400/

 

[chwala]

I'm not sure I understand what you want to know. However, I have a few interesting links:

• Theory
  https://en.wikipedia.org/wiki/Euler_method
  https://en.wikipedia.org/wiki/Heun's_method
  https://en.wikipedia.org/wiki/Runge–Kutta_methods
• Solution
  https://www.wolframalpha.com/input?i=y'=y-x
• Playground
  http://elsenaju.info/Rechner/DGL-1-Ordnung.htm
  https://planetcalc.com/8400/

Appreciated...i just wanted to check that my reasoning is correct; then i should be able to determine stability/asymptotic stability or unstability of solutions related to these fields...refreshing on this. cheers man.

and i just clicked the linear differential equation on wolfram. How did they end up with the constant $1$ on $y=c_1e^x+x+1$ in my working i have

$\dfrac{dy}{dx}-y=-x$

using integration factor,

$e^{\int({-1})dx}= \dfrac{1}{e^x}$

...
I end up with,

$ye^{-x} = \int (-xe^{-x} )dx$

...and on integration by parts on rhs... i get,

$ye^{-x}=xe^{-x}+e^{-x} +c$

$y = x+1+ce^x$ ahhhh ok boss i can see .

nice day.

 

[fresh_42]

Appreciated...i just wanted to check that my reasoning is correct; then i should be able to determine stability/asymptotic stability or unstability of solutions related to these fields...refreshing on this. cheers man.

It is similar to what we call "curve discussion" here, and Wiki "curve sketching": differentiate the function three times, determine the zeros, plug in the zeros from one derivative into the other ones, and draw the curve with the information about zeros, local minima, local maxima, asymptotes, inflection points, convex, and concave areas, etc.

It is only a bit more complicated with differential equation systems, but yes, determine as much data as possible: repellers, attractors, possible flows, and so on.

 

[WWGD]

Are you trying to plot a Vector Field; specifically here, f(x,y)=y-x, and/or its Characteristic curves?

 

[chwala]

Are you trying to plot a Vector Field; specifically here, f(x,y)=y-x, and/or its Characteristic curves?

Not really,... Just looking at how to plot directional fields...