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
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1 Answer
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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
answered May 4, 2023 at 7:33
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$\begingroup$ Sir, thank you for your answer! Your answer has helped me. And I have corrected my expression writing in the original problem. $\endgroup$ Commented May 4, 2023 at 7:42
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2$\begingroup$ Or, if higher derivatives are present
{x[t] -> 1, Derivative[n_][x][t] :> x^n}$\endgroup$– rhermansCommented May 4, 2023 at 8:10 -
$\begingroup$ @rhermansr Yep, thanks for your inspired ideas. $\endgroup$ Commented May 4, 2023 at 12:21