Suppose that $y_0$ is a solution to $y''+y'-2y=F(x)$ defined on all of $\mathbb{R}$; and that there is another solution, $y_1$ also defined on $\mathbb{R}$. Show that $|y_1(x)-y_0(x)|\to 0$ as $x\to \infty$.
Here's what I've tried:
$$|y_1-y_0|=\frac{1}{2}\left|y_0''+y_0'-y_1''-y_1'\right|\le \frac12\left(\left|y_0''-y_1''\right| + \left|y_0'-y_1'\right|\right)$$
Now why should $\lim\limits_{x\to \infty} \left|y_0''-y_1''\right|=\lim\limits_{x\to \infty} \left|y_0'-y_1'\right|=0$? I'd appreciate some hint(s).
This is not true: take $F$ constantly equal to zero; then both $0$ and $e^x$ are solutions, and their difference does not tend to zero as $x \to \infty$.