#exam 2
#u is temperature
#t is time in seconds
#x is position
#the rod is 50 cm long
j,t=var('j,t')
print('u(x,t) = (80/pi)*sum((1/(2*j+1))*exp(-(2*j+1)^2*pi^2*t/2500)*sin((2*j+1)*pi*x/50), j, 0, 10)')

j,t=var('j, t')
f(x,t)= (1/(2*j+1))*exp(-(2*j+1)^2*pi^2*t/2500)*sin((2*j+1)*pi*x/50)
f=sum(f(x,t),j,0,9)
u(x,t)= ((80/pi)*f)

#plot the graph as the temperature = 5
p1=plot(u(5, t), t, 0, 100, gridlines='minor')
p1.show()

#observations
print('Looking at the graph above we can see that the temperature starts off high and then decreases significantly fast')

#plot the graph as time = 300 sec
p2=plot(u(x, 300), x, 0, 50, gridlines='minor')
p2.show()

#observations
print('when time is 300 secs we can see that the graph above starts at 0 and is at its highest point when in the middle, it then drops back to 0')

#plot the contour plot
contour_plot(u(x,t),(x,0,50),(t,0,300), aspect_ratio=0.1,colorbar=true)

#observations
print('the graph is shaped as a curve and as you start on the outside it is hot and then as it comes closer to the middle and to the bottom the temperature is lower')