# Calculus

Suppose that we use Euler's method to approximate the solution to the differential equation
𝑑𝑦/𝑑𝑥=𝑥^4/𝑦 𝑦(0.1)=1

Let 𝑓(𝑥,𝑦)=𝑥^4/𝑦.
We let 𝑥0=0.1 and 𝑦0=1 and pick a step size ℎ=0.2. Euler's method is the the following algorithm. From 𝑥𝑛 and 𝑦𝑛, our approximations to the solution of the differential equation at the nth stage, we find the next stage by computing

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1. at (0.1 , 1 )
slope = dy/dx = (0.1)^4 /1 = (10^-1)^4 = 10^-4
at x = x + h = 0.1 + 0.2 = 0.3
Using Euler
our guess at y = 1 + h (dy/dx) = 1 + 0.2 * 10^-4 = 1 +0.00002 = 1.00002

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Damon

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