第 17 章 二階微分方程 (Second-Order Differential Equations) …

1 17 . (Second- order Di erential Equations).. 210.. 211. ( homogeneous linear Di erential equa - tions). 2. (1) P (x) dx 2 + Q (x) dx + R (x) y = G (x) . d y dy (second- order linear di erential equation), P (x) Q (x) R (x) .. (2) G (x) 0, ( homogeneous ) . 2. y1 y2 P (x) dx 2 + Q (x) dx + R (x) y = 0 , c1. d y dy c2 , c1 y1 + c2 y2 . y1 y2 , , (lin- early independent) . 2. y1 y2 P (x) dx2 + Q (x) dx + R (x) y = 0 , . d y dy , (general solutions) y = c1 y1 (x) + c2 y2 (x) . ay 00 +by 0 +cy = 0, a 6= 0 , ar2 +br+c =. 0 (characteristic equation or auxiliary equation) . (1) b2 4ac > 0, r1 r2 , y = c1 er1 x + c2 er2 x . (2) b2 4ac = 0, r , y = c1 erx + c2 xerx . (3) b2 4ac < 0, r1 = + i r2 = i , . y = e x (c1 cos x + c2 sin x).

2 Y 00 + y 0 6y = 0 . 2. 3 dx d y 2 +. dy dx y = 0 . 210. 17 . 4y 00 + 12y 0 + 9y = 0 . y 00 6y 0 + 13y = 0 . (Initial-value problems). y 00 + y 0 6y = 0 y (0) = 1 y 0 (0) = 0 . y 00 + y 0 = 0 y (0) = 2 y 0 (0) = 3 . (Boundary-value problems). y 00 + 2y 0 + y = 0 y (0) = 1 y (1) = 3 . (Nonhomogeneous linear Di erential equa - tions). ay 00 + by 0 + cy = G (x), (particular solution) yp (x) . ay 00 + by 0 + cy = 0 complementary , yc (x) . y (x) = yp (x) + yc (x) . (Method of indetermined coe cients). y 00 + y 0 2y = x2 . y 00 + 4y = e3x . y 00 + y 0 2y = sin x . y 00 + 2y 0 + 4y = x cos 3x . y 00 4y = xex + cos 2x . y 00 + y = sin x . y 00 4y 0 + 13y = e2x cos 3x . (Method of variation of parameters). ay 00 +by 0 +cy = 0 y = c1 y1 (x)+c2 y2 (x) ay 00 +by 0 +cy = G (x).

3 Yp (x) = u1 (x) y1 (x)+u2 (x) y2 (x) u01 y1 +u02 y2 = 0 , a (u01 y10 + u02 y20 ) =. G u01 , u02 , u1 (x) u2 (x) . y 00 + y = tan x 2 < x < 2 . e 2x y 00 + 4y 0 + 4y = x3.. , 211.