How do you solve this 2nd ODE for a pendulums displacement...

[applestrudle]

..when it is released from rest with velocity (v₀, 0)

I can get 1.6.5 but I can't get this:

 

[Geofleur]

First, note that $\tilde{r}(0) = A + B = 0$, because the pendulum starts off at the origin. This equation gives $A = -B$. Next, calculate the time derivative of $\tilde{r}$, evaluate it at time zero, and set it equal to $v_0 + i(0) = v_0$:

$i[(\omega_1-\Omega)A-B(\Omega+\omega_1)] = v_0$.

But the only way this equation could be true is if

$(\omega_1 - \Omega)A - B(\Omega+\omega_1)=-i v_0$.

Do you see it? The rest follows by using $A = -B$.