Let

${x}^{\prime}(t)=f(t,x(t)),t\in (0,T)$ with $x(0)={x}_{0}$

$f$ satifies the Lipschitz-condition $f(t,x)-f(t,y)\le L|x-y|$

$h\in (0,\frac{1}{L})$ is the step size and the approximation ${x}_{k}$ for $x({t}_{k})=hk$ is given by ${x}_{k}={x}_{k-1}+hf({t}_{k},{x}_{k})$.

Now I would be very interested how to derive the error

$|{x}_{k}-x({t}_{k})|\le \frac{1}{1-Lh}(|{x}_{k-1}-x({t}_{k-1})|+\frac{{h}^{2}}{2}\underset{s\in [0,T]}{max}|{x}^{\u2033}(s)|)$

I tried to look up it up in some numerical analysis books but it is always different

${x}^{\prime}(t)=f(t,x(t)),t\in (0,T)$ with $x(0)={x}_{0}$

$f$ satifies the Lipschitz-condition $f(t,x)-f(t,y)\le L|x-y|$

$h\in (0,\frac{1}{L})$ is the step size and the approximation ${x}_{k}$ for $x({t}_{k})=hk$ is given by ${x}_{k}={x}_{k-1}+hf({t}_{k},{x}_{k})$.

Now I would be very interested how to derive the error

$|{x}_{k}-x({t}_{k})|\le \frac{1}{1-Lh}(|{x}_{k-1}-x({t}_{k-1})|+\frac{{h}^{2}}{2}\underset{s\in [0,T]}{max}|{x}^{\u2033}(s)|)$

I tried to look up it up in some numerical analysis books but it is always different