Let's start by solving a single, first-order, ordinary differential equation. We'll compare the exact solution to the numerical solution:
var('x, y')
y = function('y', x)
ode = diff(y, x) + 1 / x * y == x * y^2
sol = desolve(ode, y, ics=[10, 1])
sol.show()
exact_plot = plot(sol, (x, 0.1, 10), color='red', marker='.')
rk4_plot = desolve_rk4(ode, y, ics=[10, 1], output='slope_field',
end_points=[0, 10])
show(exact_plot + rk4_plot, figsize=(4, 3))The symbolic solution is shown in the following screenshot, and its graph is plotted on the same axes as the numerical solution and the slope field:
As we explained in the previous chapter, we defined a symbolic ordinary differential equation and found the solution using desolve. The warning messages occur during the calculation of the slope field, and can be safely ignored. We then used the desolve_rk4 function to compute the solution numerically. As its name implies, the functions...