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My textbook claims that, for small step size $h$, Euler's method has a global error which is at most proportional to $h$ such that error $= C_1h$. It is then claimed that $C_1$ depends on the initial value problem, but no explanation is given as to how one finds $C_1$.

So if I know $h$, then how can I deduce $C_1$ from the IVP?

Given an IVP:
$$frac{dy}{dt}=f(t,y), y(a)=y_0, tin [a,b].$$
Here is a Theorem from Numerical Analysis by Sauer:

Assume that $f(t,y)$ has a Lipschitz constant $L$ for the variable $y$ and that the solution $y_i$ of the initial value problem at $t_i$ is approximated by $w_i$, using Euler's method. Let $M$ be an upper bound for $|y''(t)|$ on $[a,b]$. Then $$|w_i-y_i|le frac{Mh}{2L}(e^{L(t_i-a)}-1).$$

The proof is based on the following lemma:

Assume that $f(t,y)$ is Lipschitz in the variable $y$ on the set $S=[a,b]times [alpha,beta]$. If $Y(t)$ and $Z(t)$ are solutions in $S$ of the differential equation $y'=f(t,y)$ with initial conditions $Y(a)$ and $Z(a)$ respectively, then $$|Y(t)-Z(t)|le e^{L(t-a)}|Y(a)-Z(a)|.$$

Sketch of proof of the first theorem:

Let $g_i$ be the global error, $e_i$ be the local truncation error, $z_i$ satisfy the local IVP:
$$z_i'=f(t,z_i),z_i(t_i)=w_i, tin [t_i,t_{i+1}].$$

Then
$$g_i=|w_i-y_i|=|w_i-z_i(t)+z_i(t)-y_i|le |w_i-z_i(t)|+|z_i(t)-y_i|\
le e_i+e^{Lh}g_{i-1}\
le e_i+e^{Lh}(e_{i-1}+e^{Lh}g_{i-2})le cdots\
le e_i+e^{Lh}e_{i-1}+e^{2Lh}e_{i-2}+cdots +e^{(i-1)Lh}e_1.$$
Since each $e_ile frac{h^2M}{2}$, we have
$$g_ile frac{h^2M}{2}(1+e^{Lh}+cdots+e^{(i-1)Lh})=frac{h^2M(e^{iLh}-1)}{2(e^{Lh}-1)}le frac{Mh}{2L}(e^{L(t_i-a)}-1).$$

Hope this helps.

Consider the IVP $y'=f(t,y)$, $y(t_0)=y_0$. Let $t_k=t_0+kh$, $y_k$ computed by the Euler method. We know that the error order of the Euler method is one. Thus the iterates $y_k$ have an error relative to the exact solution $y(t)$ of the form $$y_k=y(t_k)+c_kh$$ with some coefficients $c_k$ that will be closer determined in the course of this answer.

Now insert this representation of $y_k$ into the Euler step and apply Taylor expansion where appropriate
begin{align}
[y(t_{k+1})+c_{k+1}h]&=[y(t_k)+c_kh]+hf(t_k,[y(t_k)+c_kh])\
&=y(t_k)+c_kh+hBigl(f(t_k,y(t_k))+h,∂_yf(t_k,y(t_k)),c_k+O(h^2)Bigr)\
y(t_k)+hy'(t_k)+tfrac12h^2y''(t_k)+O(h^3)&=y(t_k)+hy'(t_k)+hBigl[c_k+h,∂_yf(t_k,y(t_k)),c_kBigr]
end{align}
where $∂_y=frac{partial}{partial y}$ and later $∂_t=frac{partial}{partial t}$.

In the Taylor series for $y(t_{k+1})=y(t_k+h)$ on the left side the first two terms cancel against the same terms on the right side. The second derivative can be written as
$$
y''(t)=frac{d}{dt}f(t,y(t))
=∂_tf(t,y(t))+∂_yf(t,y(t)),f(t,y(t))
overset{rm Def}=Df(t,y(t)).
$$
Divide the remaining equation by $h$ and re-arrange to get a difference equation for $c_k$
$$
c_{k+1}=c_k+hBigl[∂_yf(t_k,y(t_k))c_k-tfrac12Df(t_k,y(t_k))Bigr]+O(h^2).
$$

This looks like the Euler method for the linear ODE for a continuous differentiable function $c$,
$$
c'(t)=∂_yf(t,y(t))c(t)-tfrac12Df(t,y(t)),~~text{ with }~~c(t_0)=0.
$$
Again by the first order of the Euler method, $c_k$ and $c(t_k)$ will have a difference $O(h)$, so that the error we aim to estimate is
$$y_k-y(t_k)=c(t_k)h+O(h^2).$$

Now if $L$ is a bound for $∂_yf$, the $y$-Lipschitz constant, and $M$ is a bound for $Df=∂_tf+∂_yf,f$, or the second derivative, then by Grönwall's lemma
$$
|c'|le L|c|+frac12Mimplies |c(t)|le frac{M(e^{L|t-t_0|}-1)}{2L}
$$
which reproduces the usual specific estimate of the coefficient of the error term.