Let us reduce it to a scalar linear ODE. In general,
$$f'=\left(\begin{array}{cc}A&B\\ C&D\end{array}\right)f$$ is equivalent to
$$w''+pw'+qw=0,$$
where $w$ is the first component of $f$, and $p=-(B'/B+A+D),$ and

$q=-A'+AB'/B+AD-BC$. So, if my computation is correct, we obtain
$$p=-1/t,\quad q=-(\gamma^2+\beta^2)t^2+2\alpha\beta t-\alpha^2+\alpha/t.$$
Further, on can kill $p$ by setting $w=y\sqrt{t}$ and obtain
$$y''+\left(-(\beta^2+\gamma^2)t^2+2\alpha\beta t-\alpha^2+\frac{\alpha}{t}-\frac{3}{4t^2}\right)y=0.$$
Edit. One can show that $0$ is an apparent singularity, so this equation can be reduced to the Weber equation (parabolic cylinder) $u''=(t^2+c)u$, but a simpler way to do this reduction is indicated in the answer of Michael Renardy. There are countably many $c=c_n$ for which this equation has an elementary solution.

For generic $\alpha,\beta,\gamma$, the solutions are not elementary functions..