Find the relationship between $x_0$ and $y_0$ that ensures $x(t) → 0$ as $t → ∞$ If the roots of the auxiliary equation of a second order linear homogeneous
ODE are $k_1 > 0$ and $−k_2 < 0$ then the solution is
$$x(t) = Ae^{k_1t}+ Be^{−k_2t}.$$
For most choices of initial conditions
$x(0) = x_0$, $\dot x(0) = y_0$
we will have that $x(t) → ±∞$ as $t → ∞$. However, there are some special
initial conditions for which $x(t) → 0$ as $t → ∞$. Find the relationship between
$x_0$ and $y_0$ that ensures this
My working so far is $x_0 = A + B$ and $y_0 = Ak_1 - Bk_2$ and I'm really unsure of how to tackle this question further

The second term always converges to zero for $t\to\infty$. The first term always diverges at the same time  if $A\ne 0$. Thus you need $A=0$, elimination of $B$ gives the desired relation.
