Jawab:

ℒ(y', y, x) = 1/2 (-4 e^((12 x)/5) y^2 + e^((12 x)/5) (y')^2 + 48 e^((12 x)/5) y sin(2 x))

Penjelasan dengan langkah-langkah:

Input

5 y''(x) + 12 y'(x) + 20 y(x) = 120 sin(2 x)

ODE classification

second-order linear ordinary differential equation

Alternate forms

y''(x) = -(12 y'(x))/5 - 4 y(x) + 24 sin(2 x)

5 y''(x) + 12 y'(x) + 20 y(x) = 240 sin(x) cos(x)

5 y''(x) + 12 y'(x) + 20 y(x) = 60 i e^(-2 i x) - 60 i e^(2 i x)

Differential equation solution

y(x) = c_1 e^(-(6 x)/5) sin((8 x)/5) + c_2 e^(-(6 x)/5) cos((8 x)/5) - 5 cos(2 x)

Possible Lagrangian

ℒ(y', y, x) = 1/2 (-4 e^((12 x)/5) y^2 + e^((12 x)/5) (y')^2 + 48 e^((12 x)/5) y sin(2 x))