Question

Tentukan penyelesaian umum dari persamaan:
y″ - 2y′ + 4y = 8x - 12sin(2x)

Answer 1

Gunakan transformasi Laplace.

saya anggap y(0) = a dan y'(0) = b

y'' - 2y' + 4y = 8x - 12sin(2x)
L(y'' - 2y' + 4y) = L(8x - 12sin(2x))
L(y'') - 2L(y') + 4L(y) = 8L(x) - 12L(sin(2x))
(s²f(s) - s.f(0) - f'(0)) - 2(sf(s) - f(0)) + 4f(s) = 8/s² - 12.4/(s² + 4)
s²f(s) - as - b - 2sf(s) + 2a + 4f(s) = 8/s² - 48/(s² + 4)
s²f(s) - 2sf(s) + 4f(s) = 8/s² - 48/(s² + 4) + as + b
(s² - 2s + 4)f(s) = 8/s² - 48/(s² + 4) + as + b
f(s) = (8/s² - 48/(s² + 4) + as + b)/(s² - 2s + 4)
f(s) = 8/(s²((s² - 2s + 4))) - 48/((s² - 2s + 4)(s² + 4)) + as/(s² - 2s + 4) + b/(s² - 2s + 4)