The euler function computes and displays approximations to points on the solution curve for the differential equation y'=f(t,y) with initial condition init_t, init_y. The step size is (final_t-init_t)/m. The function displays the points calculated and also returns a list of the points.
def euler(f,init_t,init_y,final_t,m):
p=lambda t,y: y+((final_t-init_t)/m)*f(t,y);
tt=init_t; yy=init_y;
data=[[tt,yy]];
for i in range(m):
yy=p(t=tt,y=yy)
tt=tt+(final_t-init_t)/m
print n(tt),n(yy)
data.append([tt,yy]);
return data
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pts=euler(lambda t,y: t-y^2,0,0,1,5)
0.200000000000000 0.000000000000000 0.400000000000000 0.0400000000000000 0.600000000000000 0.119680000000000 0.800000000000000 0.236815339520000 1.00000000000000 0.385599038513605 0.200000000000000 0.000000000000000 0.400000000000000 0.0400000000000000 0.600000000000000 0.119680000000000 0.800000000000000 0.236815339520000 1.00000000000000 0.385599038513605 |
Now we can plot the points joined by line segments.
p1=list_plot(pts,plotjoined=true)
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Plot the slope field.
dummy=var("x y"); p2=plot_slope_field(x-y^2,(x,0,1),(y,0,0.5))
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Display the two plots together.
show(p1+p2)
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