# Euler Approximation of an Ordinary Differential Equation

Please help with the following problem, providing step by step calculations in the solution.

Solve the differential equation subject to y(0)=2. An Euler approximation to y(x)=2. An Euler approximation to y(x) is given by setting h=x/h, solving the difference equation:

See attached file for equations and full problems.

With initial condition y0=2. The approximation is then y(x)=yn, and show that if n is large, this approximates y(x).

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#### Solution Preview

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Solve the differential equation

(1)

subject to y(0) = 2. An Euler approximation to y(x) is given by setting h = x/n, solving the difference equation

(2)

with initial condition y0=2. The approximation is then y(x) ~ yn. Find y_n, and show that if n is large, ...

#### Solution Summary

We show that in the limit of small step size, Euler's approximation to a given ordinary differential equation approaches the true solution. Step by step calculations are provided for various differential equations given Euler approximations and initial conditions.