Question #41591

Let $\vec{u}, \vec{v}, \vec{w}$ be a right-handed system of mutually orthogonal unit vectors in $\mathbb{R}^3$. A family of curves $C_1$ in the plane spanned by $\vec{u}$ and $\vec{v}$ is given by the position vector $\vec{r} = x\vec{u} + y\vec{v}$ satisfying the differential equation: $d\vec{r} = \left( 3x^2 [\vec{u}\ \vec{r}\ \vec{w}] \right) \vec{u} + \left( 2y [\vec{r}\ \vec{u}\ \vec{w}] \right) \vec{v}$ where $[\vec{a}\ \vec{b}\ \vec{c}]$ denotes the scalar triple product. Let $C_2$ be the orthogonal trajectory of the family $C_1$ that passes through the point with position vector $\vec{u} + 5\vec{v}$. If the point $2\vec{u} + \alpha\vec{v}$ lies on $C_2$, find the value of $\alpha^2$.