Answer to Question #151391 in Differential Equations for Supreet

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