Given continuous $f: \mathbb{R} \rightarrow \mathbb{R}$, $\begin{vmatrix}a_1 & b_1 \\ a_2 & b_2\end{vmatrix} \neq 0$ and consider the ODE
$$y'(t) = f\left(\frac{a_1 t + b_1 y + c_1}{a_2 t + b_2 y + c_2}\right)$$
I'm asked to solve the given ODE by transforming it to an homogenous ODE (in german: auf ein homogenes System zurückführen). This means that there is a function $h: \mathbb{R} \rightarrow \mathbb{R}$ such that the equation above is equivalent to
$$y'(t) = h\left(\frac{y(t)}{t}\right)$$
Currently I'm stuck and don't have a clue how to approach this. It would be enough to show that one can express the fraction in the initial equation as a function of only $\frac{y(t)}{t}$, but I can't think of a good approach and don't know how to factor in the non-zero determinant.
This equation would be homogeneous if $c_1 = c_2 = 0$. However, it is not quite homogeneous in general. What we can do is translate the variables, setting $\tilde t = t - t_0$ and $\tilde y = y - y_0$, so that $$\begin{cases}a_1 \tilde t + b_1 \tilde y = a_1 t + b_1 y + c_1 \\ a_2 \tilde t + b_2 \tilde y = a_2 t + b_2 y + c_2\end{cases}$$ In terms of $t_0$ and $y_0$ this is a system of two linear equations, and the condition on the determinant we are given guarantees its non-singularity. With respect to the translated coordinates, we get $$\frac{d}{d \tilde t} \tilde y = \frac{d}{dt} y = f\left(\frac{a_1 \tilde t + b_1 \tilde y}{a_2 \tilde t + b_2 \tilde y}\right)$$ which is homogeneous.