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Sketch a direction field for the differential equation. Then use it to sketch three solution curves.$$y^{\prime}=x-y+1$$

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Calculus 2 / BC

Chapter 9

Differential Equations

Section 2

Direction Fields and Euler's Method

Missouri State University

Baylor University

Idaho State University

Boston College

Lectures

13:37

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. An ordinary differential equation (ODE) is a differential equation containing one or more derivatives of a function and their rates of change with respect to the function itself; it can be used to model a wide variety of phenomena. Differential equations can be used to describe many phenomena in physics, including sound, heat, electrostatics, electrodynamics, fluid dynamics, elasticity, quantum mechanics, and general relativity.

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Sketch a direction field f…

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Sketch the direction field…

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$9-10$ Sketch a direction …

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We give the parential equation. We asked to find a sopilothis differential equation and to find 3 solutions. Differential equation is y prime equals x, minus y plus 1 point. Now we see that if we have some slope c in a numbers, so that y prime is equal to c and follows it 1 equal to x, minus y plus 1, or that y will be equal to x, plus 1 minus c. This is the equation of a line with slope: 1 y intercept 1 minus c, so graphing this slope field, you see particularyelloif, we take c to be 1, we'll be a slope of 1 andthe line y equals x, polar slopes on that line to be 1. So to follow the line, it was then we'll get that is 120 point thither 1 again at 11, with the 1 again at 22 and the other points and 3 in that line, and so on. We have that c is c equal to 0 and we have the line y equals x plus 1 point. All the poison is going to have slope 0, so we zone and at 12, then 23 and so on. Ye can fill slopes. 13 is 0. If we take to be to say- and we get the line y equals x plus negative 1 is going to have minus negative 1, let's will have a slip up to an keep. Ereec is equal to 3 or c is equal to negative 1. Let'S say we'll get: the line y equals x plus 2 now have downward slope at 02 negative 1. We can continue this along the line. These take cento the negative 2, his keeper. We get the line y equals x, plus 3, so very steep 1. At 0 degree and all along that line, are these steep downward slopes. I think it's kind of easy to see the pattern here so that, as we get further and further away from the line y equals x, plus 1 or slopes it steeper and steeper they're almost vertically is a very rough drawing of what the slope there looks like. Fortunately, i laps of is and cooperating right now it's going to take a little bit, but the idea is that get very steep further than you get in the line at you're. Almost verticalthis is a very rough sketch on the slope field and were asked to find 3 solutions on this slope field. So, for example, let's take solution, have initial value: 00 graphisblack. We know this is a point that 00 point and if we follow the slopes of this is merely possit, see that there is the flat now, as we approach x, x, appris infinity on the right, that's puttin very on the left. We get this credit they steeply going down. This looks a little bit like a logarithm curve. Perse shifted to the left, however, because you had any verglas inpace on the graph draw as initial solution by 0 equals 1 of gratis in gray. This contains the .01 and to follow of slopes as nearly as possible. You see that the expert is tita, those are positive, infinity and as x, approaches positive infinity, it thirds out level and then begin slope upwards, similar to our other graph. Finally, the third solution will have initial value. 0 equals negative 1 point, so we know the graphic is going to have the .0 negative 1 on you see that his expert is negative, is frequently the negative infinity as well, and i at purchase positive infinity is that's going to slower and slower more slowly unto It'S writing about the same rate as the other 2 graphs.

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