How can I automatically obtain the characteristic equation of a second-order differential equation given any expression?

Assuming a second-order differential equation is given as follows:

x''[t] + x'[t] + 16 x[t] == 0

1 Answer 1


Your equation has wrong syntax. x^\[Prime]\[Prime]) should be written like: x'' (not back ticks and no superscript). Then we can get the characteristic equation by:

eq = x''[t] + Derivative[1][x][t] + 16 x[t] == 0;
eq /. {x[t] -> 1, x'[t] -> x, x''[t] -> x^2}

16 + x + x^2 == 0
  • $\begingroup$ Sir, thank you for your answer! Your answer has helped me. And I have corrected my expression writing in the original problem. $\endgroup$
    – chen chen
    May 4 at 7:42
  • 2
    $\begingroup$ Or, if higher derivatives are present {x[t] -> 1, Derivative[n_][x][t] :> x^n} $\endgroup$
    – rhermans
    May 4 at 8:10
  • $\begingroup$ @rhermansr Yep, thanks for your inspired ideas. $\endgroup$
    – chen chen
    May 4 at 12:21

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