Thanksgiving bonus 2

[Victor Ivrii]

If we want to find a second order equations with the fundamental system of solutions $\{y_1(x),y_2(x)\}$ s.t. $W(y_1,y_2):=\left|\begin{matrix} y_1 & y_2\\ y_1' &y_2'\end{matrix}\right|\ne 0$, we write
$$
W(y,y_1,y_2):=\left|\begin{matrix}y & y_1 & y_2\\ y' &y_1' &y_2'\\ y'' &y_1'' &y_2'' \end{matrix}\right|= 0.$$

Problem.
Find a second order equation with the fundamental system of solutions $\{y_1(x),y_2(x)\}=\displaystyle{\{\frac{1}{x+1},\frac{x}{x+1}\}}$.

[Yiwei Han]

Have a good holiday!!

[Yiwei Han]

This is the corrected answer. Happy Thanksgiving!!

[Victor Ivrii]

Full credit given, but was not a very good problem: one can replace $y_2$ by $y_1+y_2=1$ coefficient at $y$ must be $0$.