## Description

We can use a combination of commands to construct a phase plane for a system. We'll create a phase plane for the system

(1)We'll create our phase plane by combining plots of the vector field and nullclines $y = (-1/3)x$ and $y = x$

## Sage Cell

#### Code

```
var('x y')
vec = vector([x + 3*y, x - y])
mag = sqrt(vec[0]^2 + vec[1]^2)
p = plot_vector_field(vec/mag, (x, -5, 5), (y, -5, 5))
p += plot((-1/3)*x, (x, -6, 6)) + plot(x, (x, -6, 6), color='red')
p.show(xmin=-5, xmax=5, ymin=-5, ymax=5)
```

## Options

#### Adding a Solution Curve

We can add a solution curve to our plot to see how a solution behaves under given initial conditions, using the `desolve_system` method. Be careful when adding a solution curve, as it may affect the bounds of the graph; you can easily remedy this by setting `xmin, xmax, ymin` or `ymax` as necessary when calling `.show()`.

While we’re at it, if we know the straightline solutions, we may add those as well:

#### Code

```
var('t')
x = function('x')(t)
y = function('y')(t)
de1 = diff(x, t) == x + 3*y
de2 = diff(y, t) == x - y
sols = desolve_system([de1, de2], vars=[x, y], ivar=t, ics=[0, 1, 3])
curve = vector([sols[0].rhs(), sols[1].rhs()])
var('x y')
vec = vector([x + 3*y, x - y])
mag = sqrt(vec[0]^2 + vec[1]^2)
p = plot_vector_field(vec/mag, (x, -5, 5), (y, -5, 5))
p += plot((-1/3)*x, (x, -6, 6)) + plot(x, (x, -6, 6), color='red')
p += parametric_plot(curve, (t, 0, 10), color='green')
p.show(xmin=-5, xmax=5, ymin=-5, ymax=5)
```

```
var('t')
x = function('x')(t)
y = function('y')(t)
de1 = diff(x, t) == x + 3*y
de2 = diff(y, t) == x - y
sols = desolve_system([de1, de2], vars=[x, y], ivar=t, ics=[0, 1, 3])
curve = vector([sols[0].rhs(), sols[1].rhs()])
var('x y')
vec = vector([x + 3*y, x - y])
mag = sqrt(vec[0]^2 + vec[1]^2)
p = plot_vector_field(vec/mag, (x, -5, 5), (y, -5, 5))
p += plot((-1/3)*x, (x, -6, 6)) + plot(x, (x, -6, 6), color='red')
p += parametric_plot(curve, (t, 0, 10), color='green')
p += plot(-x, (x, -6, 6), color='purple') + plot((1/3)*x, (x, -6, 6), color='purple')
p.show(xmin=-5, xmax=5, ymin=-5, ymax=5)
```

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Author: Thomas Judson

Date: 25 May 2020 23:10

Submitted by: Zane Corbiere