Question

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

Answer 1

$\textsf{The DE is a differential equation with}\, e^x-2e^{-x}\\\textsf{as solution}\\ \textsf{Any DE that has}\, Ae^x + Be^{-x}\textsf{as solution}\\ \textsf{also has its constituent terms}\,\, Ae^x \\ \textsf{and}\,\, Be^{-x}\,\, \textsf{to be solutions}\\\textsf{satisfying the DE}\\ \textsf{Where}\,\, A \textsf{and}\,\, B\,\,\textsf{are constants.}\\ \textsf{The auxiliary equation is an}\\ \textsf{equation with}\,\, 1, -1 \,\,\textsf{as its roots.}\\ m = -1, 1\\ \textsf{The quadratic equation is}\\ (m - 1)(m + 1) = 0\\ m^2 - 1 = 0\\ \therefore \textsf{The homogeneous differential}\\\textsf{equation is}\\ y" - y = 0$