Question

高数微分方程

求这个微分方程详细求解过程答案已公示就是求不出来急急急
m a b F 都是常数

Answer 1

S'' +(b/m)S' =(F-a)/m
let
Sp= At^2+Bt+C
Sp'=2At+B

Sp''=2A

Sp'' +(b/m)Sp' =(F-a)/m
2A +(b/m)(2At+B) =(F-a)/m
A(2b/m)t+[2A + B(b/m)] = (F-a)/m
=> A =0 , B=(F-a)/b
The aux. equation
p^2+(b/m)p=0
p=0 or p=-b/m
let
Sg = D + Ee^[(-b/m)t]
S = Sg+ Sp = Ee^[(-b/m)t]+[(F-a)/b]t + G (1)
S(0) =0
E+G =0 (2)
S'(0) =0
-(b/m)E +(F-a)/b =0
E = m(F-a)/b^2
from (2)
E+G =0
G = -m(F-a)/b^2
from (2)
S = [m(F-a)/b^2] .e^[(-b/m)t]+[(F-a)/b]t -m(F-a)/b^2