Euler Method/Examples/Arbitrary Example 1

Example of Use of Euler Method

$y' = \paren {1 - x} y + \cos x$

$\map y 0 = 1$

Then the Euler Method generates:

$y_1 = 1 + 2 h$

as an approximatiion to $\map y h$.

By definition of the Euler Method:

$y_{n + 1} = y_n + h \map f {x_n, y_n}$

where in this case:

$\map f {x_n, y_n} = \paren {1 - x_n} y_n + \cos x_n$

and:

$x_0 = 0$

$y_0 = \map y {x_0} = \map y 0 = 1$

Hence:

$\ds y_1$

$=$

$\ds y_0 + h \map f {x_0, y_0}$

$\ds $

$=$

$\ds \map y 0 + h \paren {\paren {1 - x_0} \map y 0 + \cos x_0}$

$\ds $

$=$

$\ds 1 + h \paren {\paren {1 - 0} 1 + \cos 0}$

$\ds $

$=$

$\ds 1 + h \paren {1 + 1}$

$\ds $

$=$

$\ds 1 + 2 h$

$\blacksquare$