1) sinx cos^2y dx + dy = 0
dy / cos^2y + sinx dx = 0 | cosy <> 0 {cosy = 0 -> y = П/2 + Пk, dy = 0}
tgy - cosx + C = 0
2) y' - y = e^3x
а) y' - y = 0 -> k-1 = 0 -> k = 1 -> y = C * e^x
б) C = C(x)
Сe^x + C'e^x - Ce^x = e^3x
C' = e^2x
C = e^2x / 2 + A
y = (e^2x/2 + A) * e^x = e^3x / 2 + Ae^x
в) y(x0) = y0: y(0) = 2 -> e^0 / 2 + Ae^0 = 1/2 + A = 2 -> A = 3/2
y = e^3x / 2 + 3e^x / 2
3)
y'' + 4y' + 3y = 2x + 1
y = y0 + y1
y0'' + 4y0' + 3y0 = 0
k^2 + 4k + 3 = 0
(k + 1)(k + 3) = 0
y0 = C1 * e^(-x) + C2 * e^(-3x)
y1'' + 4y1' + y1 = 2x + 1, y1 = ax + b
4a + ax + b = 2x + 1
a = 2
8 + b = 1 -> b = -7, y1 = 2x - 7
y = 2x-7 + C1 * e^(-x) + C2 * e^(-3x)