Consider the differential equation \frac {dy}{dx} = 6x, with initial condition y(0) = 2. a. Use...

Question:

Consider the differential equation {eq}\frac {dy}{dx} = 6x {/eq}, with initial condition {eq}y(0) = 2 {/eq}.

a. Use Euler's method with two steps to estimate {eq}y {/eq} when {eq}x = 1 {/eq}.

b. What is the solution to this differential equation (with the given initial condition)?

c. What is the magnitude of the error in the two Euler approximations you found?

d. By what factor should the error in these approximations change (that is, the error with two steps should be what number times the error with four)?

Euler's Method of Approximation:

Euler's method estimates the solution to a differential equation at specific input values without actually evaluating the differential equation directly. The formula to perform Euler's method is shown to be {eq}y_{n+1} = y_{n} + h \cdot y_{n}' {/eq}, where {eq}y_{n} {/eq}, {eq}h {/eq}, and {eq}y_{n}' {/eq}, are given to find the unknown {eq}y_{n+1} {/eq} value.

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Given: {eq}y' = \frac{dy}{dx} = 6x, y(0) = 2, y(1) = ? {/eq}

A. To solve this by applying the Euler's method, the formula is needed. The formula... Mathematical Models of Euler's Circuits & Euler's Paths

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Chapter 9 / Lesson 4
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The mathematical models of Euler circuits and Euler paths can be used to solve real-world problems. Learn about Euler paths and Euler circuits, then practice using them to solve three real-world practical problems.