I am looking at the implicit euler integration scheme for an equation

Question

$dx/dt = xt + 1$

How do you arrive at the second line from the first?

Answer 1

I don't think the second line does follow from the first, for with

$x_{k + 1} = x_k + h(x_{k + 1} t_{k + 1} + 1), \tag 1$

we find

$x_{k + 1} = x_k + hx_{k + 1} t_{k + 1} + h, \tag 2$

whence

$x_{k + 1} - hx_{k + 1} t_{k + 1} = x_k + h, \tag 3$

or

$x_{k + 1}(1 - ht_{k + 1}) = x_k + h; \tag 4$

thus, provided $1 - ht_{k + 1} \ne 0$,

$x_{k + 1} = (1 - ht_{k + 1})^{-1}(x_k + h) \ne x_k + h(1 - ht_{k + 1})^{-1}. \tag 5$