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Use Euler's method with step size 0.1 to estimate $y(0.5)$, where $y(x)$ is the solution of the initial-value problem. Round your answer to 3 decimal places.\ $y' = -3x + y^2$, $y(0) = -1$. $y(0.5) = $

          Use Euler's method with step size 0.1 to estimate $y(0.5)$, where $y(x)$ is the solution of the initial-value problem. Round your answer to 3 decimal places.\
$y' = -3x + y^2$, $y(0) = -1$.
$y(0.5) = $

Use Euler's method with step size 0.1 to estimate y(0.5), where y(x) is the solution of the initial-value problem. Round your answer to 3 decimal places.y' = -3x + y^2, y(0) = -1.
y(0.5) =

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Calculus: Early Transcendentals

James Stewart 8th Edition

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09/09/2023

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First, let's write down the initial-value problem: dy/dz = 9z y(0) = 0 Show more…

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Use Euler's method with step size 0.1 to estimate y(0.5), where y(x) is the solution of the initial-value problem. Round your answer to 3 decimal places. y' = -3x + y^2, y(0) = -1. y(0.5) =

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00:01 In this problem, we are given the differential equation y ' equal to minus 3x plus y squared with the initial condition y0 equal to minus 1.

00:14 And we are going to approximate y at x equal to 0 .5 using the earlier method with a step size of 0 .1.

00:26 The earlier method says yn plus 1 is equal to yn plus fn times the step size.

00:37 Here fn is just the right -hand side of this differential equation evaluated at xn and yn.

00:52 Now, n goes like 0, 1, 2, and so on.

00:58 So the initial condition corresponds to y0.

01:04 And of course, this initial value of x is x0.

01:09 So now let's make this table.

01:12 Here the n values 0, 1, 2, 3, 4, 5.

01:18 We will have the xn values 0, 0 .1, 0 .2, 0 .3, 0 .4, 0 .5.

01:27 We will have the y values.

01:29 The starting value is 0 .1.

01:33 And we will prepare a column for fn.

01:36 And we will also prepare a column for yn plus 1.

01:39 So we will find some value here and copy it over here and so on.

01:45 Okay, now the problem tells us to round our answer to 3 decimal places.

01:52 So working with a spreadsheet and setting the number of decimal places to 3, i have obtained the following values...

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