Question

Найти решение задачи Коши.

Answer 1

$y''' - y'' + y' - y = 0 \\ y = {e}^{kx} \\ {e}^{kx} ( {k}^{3} - {k}^{2} + k - 1) = 0 \\ {k}^{2} (k - 1) + (k - 1) = 0 \\ ( {k}^{2} + 1)(k - 1) = 0 \\ \\ {k}^{2} + 1 = 0 \\ k1 = k2 = + - i \\ \\ k3 = 1 \\ \\ y = C 1{e}^{x} + C2 \sin(x) + C3 \cos(x)$

общее решение

$y(0) = 0,y'(0) = 1,y''(0) = 0$

$y' = C1 {e}^{x} + C2 \cos(x) - C3 \sin(x)$

$y'' = C1 {e}^{x} - C2 \sin(x) - C3 \cos(x)$

система:

$0 = C1 + C2 \sin(0) + C3 \cos(0) \\ 1 = C1 + C2 \cos(0) - C3 \sin(0) \\ 0 = C1 - C2 \sin(0) - C3 \cos(0) \\ \\ C1 + C3 = 0 \\ C1 + C2 = 1 \\ C1 - C3 = 0 \\ \\ C3 = - C1 \\ C2 =1 - C1 \\ C1 + C1 = 0 \\ \\ C1 = 0 \\ C3 =0\\ C2 = 1$

$y = \sin(x)$

частное решение