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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 $ y' = y + xy, y(0) = 1. $

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$y(0.5) \approx 1.761639264$

Calculus 2 / BC

Chapter 9

Differential Equations

Section 2

Direction Fields and Euler's Method

Harvey Mudd College

Baylor University

University of Nottingham

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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Use Euler's method wi…

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a) Use Euler's method…

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(a) Use Euler's metho…

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given an initial value problem, we were asked to use Oilers method with a specific steps size to find the value of a solution. At some point X were given differential equation y prime equals y plus X y with initial value. Y zero equals one and step size is 0.1, and you want to find why of 0.5 approximately. So since step sizes 0.1, we're starting at X zero equals zero. Well, expect five steps. We have x 00 Why zero is going to be y over x zero, which is one x one is going to be X. You're a pulsar step size just 10.1 and why one is going to be y zero plus our step size times our function y plus x y evaluated at x zero y zero just simply one. This is equal to 1.1 x two is going to be x one plus or step size 0.2. Why to is why one plus our step size times are function y plus X y Evaluated at x one y one 1.1 plus 0.1 times 1.1 This is equal to 1.2 to 1 I x three is equal to x two plus or step size, which is 20.3. Why three z people, too? Why two You're gonna start just writing out why two instead of the full value plus our step size 0.1 times that I have are function at x two y two, which is why two plus 0.2 times why, too. This is equal to 1.36752 x for his X three plus or step size. Just point for and wife for is why three plus or step size times a function evaluated at x three y three Just why three plus 30.3 times y three which is equal to 1.54 five to nine 76 approximately. And finally we have our step five x five is equal 2.4 plus 14.5 We have that. Why of 0.5 Under this approximation is about equal to why five, which is equal to y four plus our step size times the function evaluated at X for y for so wife work plus points for why, for which is equal to approximately Yeah 1.76 16 393 And if you use the store function on a calculator or have a calculator, which saves more digits, you can get a more accurate answer than this. I think this is pretty accurate.

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