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M. EULER’S METHOD
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There are many differential equations whose solutions cannot be written as exact
algebraic equations. However, you can still solve particular equations as accurately
as you want by numerical methods.
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Euler’s method is a procedure that can be used to construct a numerical
approximation to the solution of a first order differential equation such that the
solution curve must satisfy the initial condition when x  x0 and y  y0 .
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To help you use the method you need to understand where it comes from:
Ex. 1)
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x  1 and given that the solution curve passes through  2,3 , find
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an approximate value for f  x  at x  2.1 .
a) Given f '  x  
b) Find the f  3
SUMMARY:
dy
 f  x, y  with initial value  x0 , y0  on the solution curve, the successive
dx
approximations to the solution curve can be generated using the following recursive
equation:
Given
𝑥𝑛+1 = 𝑥𝑛 + ℎ
𝑦𝑛+1 = 𝑦𝑛 + ℎ𝑓(𝑥𝑛 , 𝑦𝑛 )