This Maple sheet discusses the first quiz, Stefan's law of radiative cooling, and Taylor's series variation on Euler's method. First, the quiz. All the questions were about the differential equation y'=y, with initial condition y(0)=1. Question 1 had parts a, b, and c, asking for the result of using Euler's method with one step, two steps, and four steps, to estimate y(4). Question 2 asked for an eyeball estimate of y(1) based on a sketch of a direction field for y'=y. The Euler's method code from last sheet is reproduced here. QyQ+SSdldWxlcjNHNiJmKjYnSSJmR0YlSSN4MEdGJUkjeTBHRiVJImhHRiVJIm5HRiU2J0kieEdGJUkieUdGJUkibUdGJUkia0dGJUklaGlzdEdGJUYlRiVDKD44JDklPjglOSY+OCciIiI+OCg3IzckRjZGOT8oRiVGPEY8RiUxRjs5KEMoPjgmLTkkNiRGNUY4PkY1LCZGNUY8OSdGPD5GOCwmRjhGPComRkZGPEZMRjxGPD5GOC1JJmV2YWxmRyUqcHJvdGVjdGVkRzYjRjg+Rj43JC1JI29wR0ZTNiNGPjckRjVGOD5GOywmRjtGPEY8RjxGPkYlRiVGJUY8

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The Maple function euler3 returns a list of n+1 pairs of numbers, when given as its five inputs (rule_for_derivative,x_0,y_0,stepsize,n). The final element of the list is op(n+1,that list). Here, then, the answers were 5, 9, and 16. The first value, 5, comes from following the red line, which has equation y=x+1, all the way across from (0,1) to (4,5). The differential equation says y'=y, and initially, y=1, and for purposes of this calculation, we make believe that y' remains identically 1, all the way across. A line with slope 1 goes from height 1 when x=0, to height 5 when x=4. The second answer, 9, comes from following the red line ONLY from (0,1) to (2,3), and then saying: now that we're here, let's look at the new value of y (3, it is), and use the slope y'=3 for the second step. Two steps, each a sideways leg of 2. But the first vertical rise is 2, while the second vertical rise is 6 because now we're following a line through (2,3) with slope 3. Go sideways 2, up 6 from (2,3) to arrive at (4,9). So the answer to part (b) is 9. The answer to part (c) is 16. There are four line segments, and the junctions are (0,1), (1,2), (2,4), (3,8), and finally, (4,16). The blue and purple lines in the plot come from using ten, and finally, 100, steps instead of just 1 or 2 or 4. There is a closed-form solution to this differential equation. Like many of the simplest DE's, the function that solves the initial value problem can be expressed in elementary terms. From calculus, we know that e^x has derivative e^x, and e^0=1. From our existence and uniqueness theorem, that makes it THE solution to y'=y with y(0)=1. QyQtSSdERXBsb3RHNiI2Ji8tSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUkieUdGJTYjSSJ4R0YlRi9GLEYsL0YvOyIiISIiIi9GLDtGMiIiJUYz

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You try to follow the arrows along a line from (0,1) to wherever that takes you at the other end. I got 2.5 once, and 3.0 the other time, all the while knowing, as you now know, that the real answer was y(1)=e=2.7182818284... But, one cannot eyeball six places, or even 2. New topic: Stefan's law of radiative cooling. Suppose we have something very hot, sitting out in space. (For that matter, it could be sitting on an insulated patch of ground, exposed to the night sky.) It will cool off, entirely (in space) or predominantly (on the ground) by radiative cooling. Stefan's law says that T'=-K(T^4-A^4) where A is the ambient temperature. Not counting distant stars, the background temperature of the sky in deep space is about 3 degrees Kelvin. That's radiation that's been going since near the beginning of time (to the best of our current scientific understanding), cooling for reasons having to do with the expansion of the universe since the Big Bang. (again, and for the last time, with the qualification that this is the current best scientific understanding of the data.) When dealing with high initial temperatures, 3 might as well be absolute zero. So, to simplify matters, we'll say this. Also, to simplify further, we'll set K equal to 1. Time, then, measured not necessarily in seconds. Perhaps in years, perhaps days, whatever. Anyhow, we're now on a purely mathematical topic. What do the solutions to y'=-y^4 with y(0)=1000 look like? QyQ+SSdzdGVmYW5HNiJmKjYkSSJ4R0YlSSJ5R0YlRiU2JEkpb3BlcmF0b3JHRiVJJmFycm93R0YlRiUsJCokOSUiIiUhIiJGJUYlRiUiIiI=

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If we're going to make a sensible use of Euler's method here, we cannot take large steps. A single step of length 1 will carry us to absolute negative thousands (of degrees.) A look at the direction field for y'=-y^4 shows that the solutions cannot go negative starting at positive. QyQtSSV3aXRoRzYiNiNJKERFdG9vbHNHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiUhIiI= LUklaGVscEc2IjYjUSdERXBsb3RGJA== QyQtSSdERXBsb3RHNiI2Ji8tSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUkieUdGJTYjSSJ4R0YlRi8sJCokRiwiIiUhIiJGLC9GLzsiIiEiIiIvRi1GNUY3

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Now one of the drawbacks to Euler's method is that it takes many steps to get high accuracy. Too many, in fact, to achieve really high accuracy. There's an improvement discussed on page 31 of your text. Use Taylor's series approximations rather than straight line approximations, and extend instead along arcs of, say, parabolas. To do this, we need a second rule, for y'', in terms of x and y. Here, we do this with y'=-y^4. From y'=-y^4 we have y''=(d/dx)(-y(x)^4)=(chain rule!) -4y^3*y'=(Now use the rule for y')=-4y^3*(-y^4)=4y^7. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEsZXVsZXJ0YXlsb3JGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSomY29sb25lcTtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC8lK2V4ZWN1dGFibGVHRj1GOQ== Let's put on our code warrior hat here for a moment. We don't really need to, nor want to, write a new Maple function from scratch. We just need to modify what we've already got. The engine within euler3 is where it says 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 and now, we want a parabolic contribution as well. So, let's modify this to have a second function input, call it g, which gives y'' as a function of x and y. Then, the code itself goes m:=f(x,y);p:=g(x,y);x:=x+h;y:=y+m*h+p*h^2/2 etc. QyQ+SSxldWxlcnRheWxvckc2ImYqNihJImZHRiVJImdHRiVJI3gwR0YlSSN5MEdGJUkiaEdGJUkibkdGJTYoSSJ4R0YlSSJ5R0YlSSJtR0YlSSJwR0YlSSJrR0YlSSVoaXN0R0YlRiVGJUMoPjgkOSY+OCU5Jz44KCIiIj44KTcjNyRGOEY7PyhGJUY+Rj5GJTFGPTkpQyk+OCYtOSQ2JEY3Rjo+OCctOSVGSz5GNywmRjdGPjkoRj4+RjosKEY6Rj4qJkZIRj5GUkY+Rj4qJkZNRj5GUiIiIyNGPkZXPkY6LUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiNGOj5GQDckLUkjb3BHRmZuNiNGQDckRjdGOj5GPSwmRj1GPkY+Rj5GQEYlRiVGJUY+

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Red is the four step Euler method. Green is the 40 step Euler method. Tan is the four step Taylor's method, and blue is the 8 step Taylor's method. Note that with the Euler method, we get a polygon that converges from below to the correct value, while with Taylor's method, we get a chain of parabolic arcs that converges from above, and faster, to the correct value. Whether things converge from below, or from above, or part one way and part the other, depends on the details of the differential equation. Here, y' was negative but increasing, while y'' was positive but decreasing. That helps explain why our numerical methods under-and-overestimate the real answer as they do here.