Question

Find a linear homogeneous differential equation, whose solutions are {e^x,e^-x,e^x-2e^-x}

Answer 1

Let us solve the linear homogeneous differential equation $y''-y=0$. The characteristic equation $k^2-1=0$ has the solutions $k_1=1$ and $k_2=-1$. Therefore, the general solution is $y=C_1e^x+C_2e^{-x}$. If $C_1=1$ and $C_2=0$ we have $y=e^x$. If $C_1=0$ and $C_2=1$ we have we have $y=e^{-x}$. If $C_1=1$ and $C_2=-2$ we have $y=e^x-2e^{-x}.$

Answer: $y''-y=0$