Writing an equation in a specific form $y^2+a y+b=x^3+c x^2+d x$

Question

I am trying to analyze the following equation:

$$\alpha y^2+\beta y=\gamma x^3+\Delta x^2+\kappa x+\delta\tag1$$

Where all the coefficients $\alpha,\beta,\gamma,\Delta,\kappa,\delta$ are integers. $\beta,\Delta,\kappa,\delta$ can be equal to zero. And $\alpha,\gamma$ will be bigger or less than zero.

Now, how can I write equation $(1)$ into the following form:

$$y^2+a y+b=x^3+c x^2+d x\tag2$$

Where all the coefficients $a,b,c,d$ all have to be integers.

Answer 1

Multiply the equation by $\alpha^3 \gamma^2$ \begin{eqnarray*} \alpha^4 \gamma^2 y^2+\alpha^3 \gamma^2 \beta y - \alpha^3 \gamma^2 \delta=\alpha^3 \gamma^3 x^3+\alpha^3 \gamma^2 \Delta x^2+ \alpha^3 \gamma^2 \kappa x. \end{eqnarray*} Now let $Y= \alpha^2 \gamma y$ and $X=\alpha \gamma x$
\begin{eqnarray*} Y^2+\alpha \gamma \beta Y - \alpha^3 \gamma^2 \delta=X^3+\alpha \Delta X^2+ \alpha^2 \gamma \kappa X. \end{eqnarray*}