Question

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

Answer 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