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Calculus 1 The differential equation dy/dx equals x2/y2 I produces a slope field with horizontal tangents at y = 2 II produces a slope field with vertical tangents at y = 2 III produces a slope field with columns of parallel segments You can view more similar questions or ask a new question71 Slope Fields and Euler's Method Def An equation involving a derivative is called a differential equation The order of a differential equation is the order of the highest derivative involved in the equation Find all functions y that satisfy dy sec 2 52 xx dx Ans y = tan x x2 5x C General Solution Initial Value ProblemSketch slope field for y' = e^x y How do the solutions behave as x grows?

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Y=e^x slope field-Consider the differential equation given by dy/dx = xy/2 A Let y=f(x) be the particular solution to the given differential equation with the initial condition Based on the slope field, how does the value of f(02) compare to calculusSketch slope field for y' = x^2 Sketch slope field

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A) y = 1− x2 d) y = cosx b) y = 1−x2 e) y = 1 1x2 c) y = ex 2Slope field of y ′ = x y with a graph of solutions satisfying , y ( 0) = 02, , y ( 0) = 0, and y ( 0) = − 02 🔗 By looking at the slope field we get a lot of information about the behavior of solutions without having to solve the equationFree slope calculator find the slope of a line given two points, a function or the intercept stepbystep This website uses cookies to ensure you get the best experience By using this website, you agree to our Cookie Policy

Each segment has the same slope as the solution curve through (x, y) and so is tangent to the curve there The resulting picture is called a slope field(or direction field) and gives a visualization of the general shape of the solution curves Figure 162a shows a slope field, with a particular solution sketched into it in Figure 162b I got y = e^(x^3/3) 2 Calculus!!The exact solution to the IVP, which is found with separation of variables, is y = e x 2 / 2 so the exact value of y(1) is e 1/2 ≈ The smaller value of h, therefore, yields a better approximation We graph the approximations obtained with h = 01 and h = 005 as well as the graph of y = e x 2 / 2 in Figure 226

Given, as given equation is not complete soo, it may be like (dy/dx) = y^2 /x or dy/dx y^2 /x = 0 1 if we take first condition , ie dy/dx = y^2 /x dy/y^2 = dx/ x take integration on both side integral{dy/y^2} = integral {dx/x} 1/y = log x logSee the answer See the answer See the answer done loading a) Sketch the slope field for y'=y/x along with several solution curves Expert Answer Who are the experts?(A) y=x2 (B) y=ex y=e−x (D) y=cosx (E) y=lnx 2 At the right is a slope field

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 The solution to the ODE y ′ = y − x is y = c e x x 1, however, this is inconsistent with the slope field that I have created Consider the curve y − x = C The slopes of the tangent lines of a solution along this curve are all y ′ = ( x C) ′ = 1, and each point in the x, y − plane is on a line of the form y = x C for some C In our case f (x,y) = x 4 (3 −y) The "computational way" to draw the slope field is to do what a simple common plotting software would do make the infinitesimal quantities not infinitesimal and turn the equation to Δy Δx = x 4 (3 − y) Then substitute some values and plot the results of the computations This discretization processCan you guess a particular solution by looking at the slope field?

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(a) Find a slope field whose integral curve through (0,0) satisfies x e^{y}y e^{x}=0 by differentiating this equation implicitly (b) Prove that if y(x) is an Join our free STEM summer bootcamps taught by expertsThe slope field of a de is based on the fact that the de can be interpreted as a statement about the slopes of its solution curves EXAMPLE 1 The de tells us that at any point (x, y) on a solution curve the slope of the curve is equal to its ycoordinate Since the de says that y is a function whose derivative is also y, we know that y = e xThe slope field for a certain differential equation shown below Which of the following could be a solution to the differential equation with the initial condition y(0) = 1?

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A direction field (slope field) is a mathematical object used to graphically represent solutions to a firstorder differential equation At each point in a direction field, a line segment appears whose slope is equal to the slope of a solution to the differential equation passing through that pointD y d x = x − y x 2 y y ( 0) = 1 Note that solving this exact equation will lead to an implicit formula for y which will have two solutions for x = 2 However, the initial value problem has a unique solution and so just one of the two solutions produced from the implicit formula will be the actual value of yDirection Field The slope field of a differential equation {eq}\displaystyle y'(x) = f(x,y) {/eq} shows the direction of the solution {eq}\displaystyle y(x) {/eq}

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Here's a slope field for the DE with superimposed plots of for (blue), (red), and (cyan) Here are the details for simplifying WolframAlpha's solution to get Mathematica's solution We use the recurrence relation With , we have y E) x y F) x y G) x y H) x y 7 y 1 dy dx 8 y x dy dx 9 yx2 dy dx 10 1 3 6 y x dy dx 11 2 1 y x dy dx 12 y sin( )x dy dx 13 y cos( )x dy dx 14 y ln x dy dx Show work on a separate piece of paper Integrate the 3 equations that it is possible to integrate Match the slope field with their differential equations A)162 Slope Fields X Y 0 0 1 1 2 2 3 3 4 4 Figure A slope field for y′(x) = 1 16 x 9− y2, along with four curves sketched "parallel" to the field 5

Slope Fields

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Calculus, Differential Equation A direction field (or slope field / vector field) is a picture of the general solution to a first order differential equation with the form Edit the gradient function in the input box at the top The function you input will be shown in blue underneath as The Density slider controls the number of vector linesSlope fields (also called vector fields or direction fields) are a tool to graphically obtain the solutions to a first order differential equationConsider the following example The slope, y'(x), of the solutions y(x), is determined once we know the values for x and y, eg, if x=1 and y=1, then the slope of the solution y(x) passing through the point (1,1) will be A slope field or tangent fields is a graph that shows a short line segmernt with slope f(x,y) at every point to the differential equation \( y' = f(x,y) \) in a given range Plotting such line segments is very tiresome to do by hand, so learning how to do this with a computer algebra system is incredibly useful

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 Section 12 Direction Fields This topic is given its own section for a couple of reasons First, understanding direction fields and what they tell us about a differential equation and its solution is important and can be introduced without any knowledge of how to solve a differential equation and so can be done here before we get into solving themSlope Field Program (TI›84//) This program draws the slope eld for the differential equation dy dx To get rid of previously drawn slope eld, press ClrDraw (from the 2nd DRAW menu) or change Your solution curve should resemble y= exfor x 0 381EXAMPLE 6 Constructing a Slope Field Construct a slope field for the differential equation dy dx cos x SOLUTION We know that the slope at any point (0, y) will be cos 0 1, so we can start by drawing tiny segments with slope 1 at several points along the yaxis (Figure 62a) Then, since the slope at any point (p, y) or ( p, y) will be 1, we

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AP Slope Fields Worksheet Key S Stirling 1112 Page 1 of 7 AP Slope Fields Worksheet Slope fields give us a great way to visualize a family of antiderivatives, solutions of differential equations Solving 2 dy x dx = means "Name a function whose derivative is 2x" Answers might includey =x2, yx=2 3, yx= 2 −4, and so forth In generalTextbook solution for Calculus (MindTap Course List) 11th Edition Ron Larson Chapter Problem 41E We have stepbystep solutions for your textbooks written by Bartleby experts! Then supposing h(0) = 100 meters, how long does it take for the ball to hit the ground Exercise 1E 1112 Find the general solution of y ′ = ex, and then y ′ = ey Exercise 1E 1113 Solve dy dx = ex x and y(0) = 10 Answer y = e x x 2 2 9 Exercise 1E 1114 Solve x ′

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Question a) Sketch the slope field for y'=y/x along with several solution curves This problem has been solved! Slope is given by dy dx = − e−x Answer linkSlope Field Generator Log InorSign Up Let g(x,y)=dy/dx 1 g x, y = 2 x y

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However, we could use Mathematica to superimpose its graph on the slope field and compare If we do this, the result would look something like the following Slope Field for dy/dx = x y with superimposed solution curves y = A e x 2 /2 As you can see, the solution curves that we just calculated theoretically fit the slope field that we made earlier quite wellA Slope Field is a graphical general solution to a differential equation A Particular Solution in a Slope Field is a Continuous Function passing through an initial condition Recall that a derivative function gives us the slopes of a function at a point When we work with (A ) y x 2 (B ) y e x (C )Through the slope field going both left and right from the starting point (0, –1) Slope Fields on the AP Exams The availability of technology to draw slope fields is relatively new Some textbooks do not mention slope fields, so this is a topic that may need supplementing Graphing calculators and programs like Winplot will draw slope fields

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In words, the differential equation says "the slope at any point in the slope field is the sum of the x and half of the ycoordinate at that point" Use a lattice with x going from –3 to 3 and with y doing the same thing Start at x=3 x y dy/dx Action to take 3 3 33/2, or 45 Draw short segment with slope 45 at (3,3)1842 Graph the slope field and the solution for the initialvalue problem y' = e –x 2 and y(3) = 001 Use a 3, 3 x 05, 2 window Remember to use t in place of x in the Y = Editor Click here for the answer The Slope Field of y' = y The slope field for the differential equation y' = y is The idea behind a direction field is the fact that the derivative of a function evaluated at a given point is the slope of the tangent line to the graph of that function at the same point Other examples of differential equations for which we can create a direction field include y ′ = 3x 2y − 4 y ′ = x2 − y2

Slope Fields

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Slope Fields Nancy Stephenson Clements High School Sugar Land, Texas Draw a slope field for each of the following differential equations Each tick mark is one unit 1 1 dy x dx = 2 2 dy y dx = 3 dy x y dx = 4 2 dy x dx = 5 1 dy y dx =− 6 dy y dx x =− Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields It only takes a minute to sign up Ce C y e x y x y The calculator drawn slope field for the differential equation dy x y dx is shown in the figure below a) Sketch the solution curve b) Sketch the solution curve passing through the point (0,1) passing through the point (3,0)

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Slope field to explain why _____ 18 Consider the differential equation given by dy x dx y = (a) On the axes provided, sketch a slope field for the given differential equation (b) Sketch a solution curve that passes through the point (0, 1) on your slope field (c) Find the particular solution y = f (x) to the differential equation with theSlope Fields and Differential Equations Students should be able to Draw a slope field at a specified number of points by hand Sketch a solution that passes through a given point on a slope field Match a slope field to its differential equation Match a slope field to its solutionC) Plot the estimated solution on the slope field ;compare the solution and the slope field d) Check that {eq}y = e^x {/eq} is the solution of {eq}y ' = y {/eq} with {eq}y (0) =1 {/eq} Euler's

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Slope Fields (Varsha Gopal) Draw the slope field of the following equations 3 dx dy =x y 4 dx dy =x 2 33Find a general solution for y ′′y ′=e x 34Use the Wronskian to show that y 1=√x and y 2=x −1 are particular solutions of the differential equation 2x 2yGiven any point (x,y) we can use this to find the slope of our solution at that point For the above example, the slope of the solution at the point (3,2) is 2 4 y' = = 2 3 We can think of the slope of the direction of the flow at that point From the point (3,2), we can draw a small line segment with slopeSlope Field Worksheet #2 Name _____ 1 At the right is a slope field from a certain differential equation Which of the following could be a specific solution to that differential equation?

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