Use a CAS to explore graphically each of the differential...

Use a CAS to explore graphically each of the differential equations. Perform the following steps to help with your explorations. a. Plot a slope field for the differential equation in the given $x y$-window. b. Find the general solution of the differential equation using your CAS DE solver. c. Graph the solutions for the values of the arbitrary constant $C=-2,-1,0,1,2$ superimposed on your slope field plot. d. Find and graph the solution that satisfies the specified initial condition over the interval $[0, b].$ e. Find the Euler numerical approximation to the solution of the initial value problem with 4 subintervals of the $x$-interval and plot the Euler approximation superimposed on the graph produced in part (d). f. Repeat part (e) for $8,16,$ and 32 subintervals. Plot these three Euler approximations superimposed on the graph from part (e). g. Find the error $(y$ (exact) $-y$ (Euler)) at the specified point $x=b$ for each of your four Euler approximations. Discuss the improvement in the percentage error. $y^{\prime}=x+y, \quad y(0)=-7 / 10 ; \quad-4 \leq x \leq 4, \quad-4 \leq y \leq 4;$ $b=1$

Use a CAS to explore graphically each of the differential equations. Perform the following steps to help with your explorations.
a. Plot a slope field for the differential equation in the given $x y$-window.
b. Find the general solution of the differential equation using your CAS DE solver.
c. Graph the solutions for the values of the arbitrary constant $C=-2,-1,0,1,2$ superimposed on your slope field plot.
d. Find and graph the solution that satisfies the specified initial condition over the interval $[0, b].$
e. Find the Euler numerical approximation to the solution of the initial value problem with 4 subintervals of the $x$-interval and plot the Euler approximation superimposed on the graph produced in part (d).
f. Repeat part (e) for $8,16,$ and 32 subintervals. Plot these three Euler approximations superimposed on the graph from part (e).
g. Find the error $(y$ (exact) $-y$ (Euler)) at the specified point $x=b$ for each of your four Euler approximations. Discuss the improvement in the percentage error.

$y^{\prime}=x+y, \quad y(0)=-7 / 10 ; \quad-4 \leq x \leq 4, \quad-4 \leq y \leq 4;$
$b=1$

Key Concepts

Error Analysis in Numerical Methods

Error analysis involves comparing numerical approximations with exact solutions to assess the accuracy of numerical methods. By computing percentage errors or absolute differences at specific points, one can gauge the effectiveness of the numerical approach and understand how reducing the step size or refining the method leads to improved approximation accuracy.

Initial Value Problems (IVPs)

An initial value problem is a type of differential equation that is accompanied by specified values (initial conditions) of the solution at a given point. These conditions allow for the determination of the arbitrary constant in the general solution, thereby yielding a unique, specific solution that satisfies both the differential equation and the initial condition.

Euler's Method

Euler's method is a numerical approach for approximating solutions to initial value problems for differential equations. It approximates the solution by stepping forward in small increments along the independent variable and using the differential equation’s slope to estimate changes in the function value. The accuracy of Euler's method improves with smaller step sizes.

Slope Fields (Direction Fields)

A slope field, or direction field, is a graphical tool for visualizing the behavior of differential equations without solving them explicitly. It consists of short line segments drawn at selected points in the domain, where each segment’s slope corresponds to the value of the derivative at that point, thereby providing an overall sense of the solution curves' orientations.

Ordinary Differential Equations (ODEs)

An ordinary differential equation (ODE) is an equation involving a function and its derivatives with respect to one independent variable. ODEs describe various dynamic behaviors in systems by relating the rate of change of a quantity to the quantity itself or to other variables, forming the mathematical basis for many models in science and engineering.

General Solution of a Differential Equation

The general solution of a differential equation represents the complete set of all possible solutions and includes an arbitrary constant. It is obtained through integration techniques and encapsulates every potential solution, from which specific solutions can be derived by applying initial or boundary conditions.

Transcript

00:01 Okay, what we're going to do is we're going to step through the process of doing multiple things.

00:07 So we're going to start with a differential equation, y prime, is equal to x plus y.

00:16 And the first thing we're going to do is we want to plot the slope field.

00:25 And we want to do it over the x interval of negative 4.

00:31 To positive 4 and the y interval of negative 4 to positive 4 as well and so um we're going to go ahead i'm going to switch to um a slope field generator which is actually desmos um and so i have slope field generator by desmos i already have x plus y um in my um different equation place.

01:03 So here is the slope field.

01:07 And you can zoom in if you want to.

01:10 I'm going to keep it from basically negative 6 to 6, but you can zoom in if you would like to.

01:18 It's a little bit harder with desmos, but if you have a tia inspire, you can set your window as well.

01:25 So that's the first thing.

01:26 So this is the slope field of x, y prime equal to x plus y.

01:32 And then the second thing we want to do is we want to find the general solution.

01:44 Okay, so what i'm going to do is i'm going to switch to a differential equation solver.

01:50 This happens to be symbilab.

01:54 And so as you can see, i've already put, and there's multiple ones on the, on the internet, y prime is equal to x plus y.

02:01 And so when i did that, this one, since it's free, you're going to have to deal with some advertisements.

02:10 And then here is the solution right here.

02:14 And then below it, you can also see showing the steps of how the symbolab actually calculated that general solution.

02:24 So here's our general solution, negative x minus 1 plus some constant number times x.

02:31 E to the x.

02:32 So there is our general solution of, let's see, y equal to negative x minus one plus some constant number times e to the x.

02:50 And then the second, the third thing we want to do is we want to graph on our slope field each solution for c1 equal to negative.

03:07 Negative 2, negative 1, 0, 1, and 2.

03:12 So that's what the third thing we want to do.

03:15 So i'm going to switch back to my slope field generator.

03:19 And let's see if we can get these in.

03:22 And so i have y equal to negative x minus, oops, that's not what i wanted to do.

03:37 So, oh, there we go.

03:40 Negative x minus 1 and then we had oh no it's going to be a minus 2 minus 2 is the first 1 and then e raised to the x so there is the first solution graft then we're going to do y equals negative x minus 1 minus 1 e raised to the x and so there is the second solution and then we're going keep going and the next one is just negative x minus one goes c1 is zero and then we're going to do y equals minus one and then it was a plus you raise the x there is so i have and as you can see these are all color -coded and then the last one is negative x minus one plus two, you to the x.

05:06 And so we'll let that pop generate.

05:09 And there we have the five particular solutions grafts in our slope field.

05:19 Okay.

05:19 And then, of course, you can always take a snapshot.

05:24 Okay.

05:25 Now what we want to do is we want to find and graph the solution with our initial condition of y of zero is equal to negative seven tenths.

05:47 So we want to know what that solution is.

05:51 And so we have negative seven tenths is equal to negative seven tenths is equal to negative.

05:59 Negative 1 plus c1e raised to the x oh raised to the zero i'm sorry because x is zero and so this is going to give me negative nope we're going to add the one over so it's three -tenth is equal to c1 so that particular solution is y is equal to negative x minus 1 plus three -tenths e to the x okay so there we have that one and so and we're also going to graph the solution that satisfies this specified initial condition over the interval over the interval from zero to one this time now what i'm going to go ahead and do is i'm going to go ahead and not do that yet because i want to be able to graph in these for a bunch of next another steps.

07:22 So we have a bunch of next another steps.

07:25 And so and what we want to do is since it's over that interval, i actually need to know what he is at 1.

07:38 So we want to add 1.

07:41 And so if i put in 1 for x, we're going to get a negative 1 .18.

07:48 So i need to keep that in mind.

07:50 Okay.

07:51 So now what we want to do is we want to find the euler approximations.

07:59 So we're going to keep going on this.

08:00 And we're going to find the euler approximations with several different sub intervals.

08:17 So the first time is four sub intervals.

08:25 And then we're going to do eight and then 16 and then 32.

08:32 And then we're going to actually superimpose each of these onto our slope field.

08:39 Okay, so that's something to remember.

08:41 And so what we're going to do is i'm going to go ahead and switch to an online uler approximation calculator.

08:49 And so i have one here.

08:50 It's through planet calc.

08:54 And so how you do this is you put in your differential equation.

09:00 Actually, you go up here and put in the differential equation.

09:03 You put in the differential equation, the initial x and y values, the point of approximation, which is that b value, and, of course, your step size.

09:14 And so the first time we want to do it is at four sub intervals, which would be 0 .25.

09:25 And then, of course, you evaluate your exact solution, which is that negative x minus 1 plus 0 .3 or 3 tenths, e raised the x.

09:35 And then you hit calculate.

09:39 And so we're going to write down a couple things.

09:42 So as i scroll on down, one, the approximation appears...

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