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

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).

17.1 齊次線性微分方程(Homogeneous Linear Differential Equa-tions) 定義 17.1.1. (1) 形如 P (x) d2y dx2 +Q(x) dy dx +R(x)y = G(x) 之微分方程稱為二階線性微分 方程(second-order linear differential equation), 其中要求P (x)、 Q(x) 和 R(x) 均為 連續函數。 (2) 若G(x) · 0, 則此微分方程 …

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  Linear, Into, Order, Differential, Homogeneous, Equa, Homogeneous linear differential equa tions

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