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Euler's Method
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Estimating numerical solutions to differential equations using Euler's Method from a known point and step size h, with successive updates xn+1=xn+h and yn+1=yn+hf(xn,yn), and interpreting the method using slope fields.
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The Idea Behind Euler's Method
AHL AI 5.16
When we have a differential equation of the form dxdy=f(x,y), Euler's Method enables us to estimate the value of y for a specific x by starting with a known point and taking small steps towards the x-value we are interested in.
It works like this:
Start at the known point.
Find the slope at the current point.
Take a horizontal step of size h
Take a vertical step of size h times the slope found in 2.
Repeat 2-5 until the desired x-value is reached.
This is useful because in the physical world, we often know the state in which a system starts, and we have equations that model how the system will change.
It is very common for these physical models not to have exact solutions, so we need to use numerical methods, of which Euler's Method is one example.
Performing Euler's Method
AHL AI 5.16
Euler's Method is a technique for approximating numerical solutions to differential equations by taking small steps in the x and y directions in accordance with the differential equation.
Mathematically Euler's Method works as follows:
Start at a known point (x0,y0)
Pick a step size h such that x0+nh=xfinal for some integer n.
Repeat the following steps for each n until the desired x-value is reached:
Find the slope dxdy=f(xn,yn)
Find the next x value xn+1=xn+h📖.
Find the next y-value yn+1=yn+h×f(xn,yn)📖
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