$$\begin{vmatrix} \alpha & x&x&x\\ x& \beta &x &x \\ x& x & \gamma &x \\ x& x& x &\delta \end{vmatrix} = f(x)-xf'(x)$$ Here's how I approached it$$ \begin{vmatrix} \alpha & x&x&x\\ x& \beta &x &x \\ x& x & \gamma &x \\ x& x& x &\delta \end{vmatrix} = g(x)$$ $$ f(x)=y$$
$$g(x)=y-x\frac{dy}{dx}$$
$$\frac{dy}{dx} -\frac{y}{x} = -\frac{g(x)}{x}$$ $$I.F = \frac{1}{x} $$
$$\frac{y}{x}=\int -\frac{g(x)}{x^2}dx$$
$$y =-x\int \begin{vmatrix}
\frac{\alpha}{x} &1 &1 &1 \\
1 &\frac{\beta}{x} &1 &1 \\
x& x & \gamma &x \\
x&x &x & \delta
\end{vmatrix} dx$$
This is where I get stuck. Computing this determinant is annoying in itself but the constant of integration is also a pain. We cannot obtain constant of integration by inputting $x=0$ in original equation and thus need to use another value of x and need to compute both $f(x)$ and $f'(x)$ for finding the constant.
Is there an easier way to solve this (by hand)?
