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}#