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Input:

w\'\'(x) + w\'(x) + w(x) = 0


ODE classification:

second-order linear ordinary differential equation


Alternate form:

w\'\'(x) = -w\'(x) - w(x)


Differential equation solution:

w(x) = c_1 e^(-x\/2) sin((sqrt(3) x)\/2) + c_2 e^(-x\/2) cos((sqrt(3) x)\/2)


Possible Lagrangian:

(script capital l)(w\', w, x) = 1\/2 (e^x w\'^2 - w^2 e^x)

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