I am reading this paper by Liniger and Willoughby on Integrating Stiff ODE. I am hoping that I am providing enough information for an answer. We are looking at an system,
$\dot{x} = -\lambda x$
The paper breaks down the integration as follows, $x(t+h) - x(t) - h [(1 - \mu) \dot{x}(t+h) + \mu \dot{x} (t)] = e_1(t)$,
where
$e_1 = - h^2 \int_0^1(\theta - \mu) \ddot{x}(t + \theta h) d\theta$
I have no idea how this came about, if someone could clue me in I would much appreciate it. I tried both the integral and Legrange remainder theorems, not sure what I am doing wrong but I could not arrive at the result.
Consider for $t$ and $h$ fixed the function $$ g(θ)=x(t+θh)-h(θ-μ)\dot x(t+θh) $$ Then \begin{align} e_1(t)=g(1)-g(0)&=\int_0^1g'(θ)\,dθ\\ &=\int_0^1\left[h\dot x(t+θh) -h\dot x(t+θh)-h^2(θ-μ)\ddot x(t+θh)\right]\,dθ\\ &=-h^2\int_0^1(θ-μ)\ddot x(t+θh) \,dθ \end{align} as claimed.