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Notes on Diffy Qs - Sage demos for section 1.2

Press the Evaluate button below to launch the Sage demonstration. You may have to wait a little before the graph appears. Be patient. To change the function, you should edit the code and press the Evaluate button again. Note that Sage ignores any line starting with a pound sign #, so those lines are simply comments for a human.

Slope field

Here we plot the slope field of $y^{\prime }=f(x,y)$ where you can change the $f(x,y)$ in the code, several possibilities are given. The default range is -2 to 2 in both x and y and you can change this as well by modifying the given variables.

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var("x y")

​

# We will plot the slope field of y' = f(x,y)

f(x,y) = 0.5*cos(x-y)-sin(x^2+y)

# Some more examples

#f(x,y) = sin(y^2)-0.5*x

#f(x,y) = cos(x*y^2)

#f(x,y) = 0.5*x

#f(x,y) = 0.5*y

#f(x,y) = 0.5*y^2

​

# The plot window

xmin = -2

xmax = 2

ymin = -2

ymax = 2

​

# Plot the slope field

sf = plot_slope_field(f(x,y), (x, xmin, xmax), (y, ymin, ymax), aspect_ratio=1, plot_points=20)

sf.show()

If you do not need a fixed aspect ratio (x and y units are the same), remove the "aspect_ratio=1," from the above. If you want to plot more little lines, you can change the "plot_points=20" parameter.

Slope field and solution

In this demonstration we plot the slope field and have the compute draw a solution with a certain initial condition $y(x_0)=y_0.$ The initial point is drawn as a red dot. Again you can change both the function and initial condition by editing the code and evaluating.

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var("x y")

​

# We will plot the slope field of y' = f(x,y)

f(x,y) = 0.5*cos(x-y)-sin(x^2+y)

# Some more examples

#f(x,y) = sin(y^2)-0.5*x

#f(x,y) = cos(x*y^2)

#f(x,y) = 0.5*x

#f(x,y) = 0.5*y

#f(x,y) = 0.5*y^2

​

# The initial condition y(x0) = y0

x0 = 0.0

y0 = 0.5

​

# The plot window

xmin = -2

xmax = 2

ymin = -2

ymax = 2

​

# Plot the slope field

sf = plot_slope_field(f(x,y), (x, xmin, xmax), (y, ymin, ymax), aspect_ratio=1, plot_points=20)

​

# Plot the solution

sol = desolve_rk4(f(x, y), y, ivar=x, ics=[x0, y0],

                   end_points=[xmin,xmax], step=0.1, output='plot')

​

# The point of the initial condition

pt = point([x0,y0], size=20, color='red')

​

# Show both plots

(sf+sol+pt).show()