Toronto Math Forum
MAT2442018F => MAT244Tests => Thanksgiving Bonus => Topic started by: Victor Ivrii on October 05, 2018, 05:33:08 PM

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 $\{y_1(x),y_2(x)\}=\displaystyle{\{\frac{1}{x+1},\frac{x}{x+1}\}}$.

Have a good holiday!!

This is the corrected answer. Happy Thanksgiving!!

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 $0$.