Question

Ребятки!!! ПОМОГИТЕ!!!

Answer 1

Ответ:

везде замена

$y = {e}^{kx}$

1.

$y'' + y' - 6y = 0 \\ {k}^{2} + k - 6 = 0\\ D = 1 + 24 = 25\\ k_1 = \frac{ - 1 + 5}{2} = 2 \\ k_2 = - 3 \\ y = C_1 {e}^{2x} + C_2 {e}^{ - 3x}$

2.

$y ''+ 4y' + 4y = 0 \\ {k}^{2} + 4 k + 4 = 0 \\ {(k + 2)}^{2} = 0 \\ k_1 = k_2 = - 2 \\ y = C_1 {e}^{ - 2x} + C_2 {e}^{ - 2x} x$

3.

$y ''- 4y' + 8y = 0 \\ {k}^{2} - 4 k + 8 = 0 \\ D = 16 - 32 = - 16\\ k_1 = \frac{4 + \sqrt{ - 16} }{2} = \frac{4 + 4i}{2} = 2 + 2i \\ k_2 = 2 - 2i \\ y = {e}^{2x} (C_1 \sin(2x) + C_2 \cos(2x) )$

общие решения