Question #656be

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Answer 1 · 2016-07-13T17:24:09

$y = C_1e^{lambda_1 x}+C_2e^{lambda_2 x}$ with
$lambda_1 =-1/2 (2 + sqrt[3])$ and
$lambda_2=-1/2(2-sqrt(3))$

This is a linear homogeneous differential equation with constant coefficients. For this equation, the solution has the structure

$y = Ce^{lambda x}$

substituting we have

$(2 lambda^2+4lambda+1/2)Ce^{lambda x} = 0$

but $Ce^{lambda x} ne 0$ so the feasible $lambda$ 's are the solutions of

$2 lambda^2+4lambda+1/2=0$

which have as solutions

$lambda_1 =-1/2 (2 + sqrt[3])$ and
$lambda_2=-1/2(2-sqrt(3))$

then, the general solution is

$y = C_1e^{lambda_1 x}+C_2e^{lambda_2 x}$