工程數學基本題

Question

求通解,用代換法檢查答案
1. y''-6y'-7y=0

2. y''-y'+2.5y=0

3 y''+y9拍平方=0

Answer 1

1.
(1)輔助(特徵)方程式 t² - 6t -7 =0 => t=7, -1
通解 y=A exp(7x) + B exp(-x)
(2)Check
y = A e^(7x) + B e^(-x)
y'=7A e^(7x) - B e^(-x)
y"=49A e^(7x) + B e^(-x)
故 y"- 6y' - 7y = A e^(7x) (49 - 6*7 - 7) + B e^(-x) (1 +6 -7)=0 (符合)

2.
(1)輔助方程式 t² - t - 2.5 =0 => t = 0.5 ± 1.5 i
通解 y = exp(x/2) * [ Asin(3x/2) + Bcos(3x/2) ]
(2)Check
y = exp(x/2) [ Asin(3x/2) + Bcos(3x/2) ]
y'= exp(x/2) [ (A-3B)/2 sin(3x/2) + (3A+B)/2 cos(3x/2) ]
y"=exp(x/2) [ (-4A-3B)/2 sin(3x/2) + (3A-4B)/2 cos(3x/2) ]
代入y''-y'+2.5y得0 (符合)

3.題目是y"+9π² y = 0 吧!
(1)輔助方程式 t² + 9π² =0 => t= 3πi, -3πi
通解為 y=A sin(3πx) + B cos(3πx)
(2)Check
y = A sin(3πx) + B cos(3πx)
y"= -9π²A sin(3πx) - 9π²B cos(3πx)
故 y"+ 9π² y=0 (符合)