求解微分方程..y''(t)+3y'(t)+y(t)=3costy(0)=0y'(0)=1

Question

求解微分方程..y''(t)+3y'(t)+y(t)=3cost y(0)=0 y'(0)=1

Answer 1

y''+3y'+y=3cost
y''+3y'+y=0
特征方程 r^2+3r+1=0
r1=(-3+√5)/2 r2=(-3-√5)/2
y=C1e^[(-3+√5)t/2]+C2e^[(-3-√5)t/2]
设y =acost + bsint
3y'=-3asint +3bcost
y'' =-acost -bsint
y''+3y'+y=(-a+3b+a)cost +(-b-3a+b)sint
y''+3y'+y=3cost
-3a=0, 3b=3,b=1
y=cost
y''+3y'+y=3cost
通解y=C1e^[(-3+√5)t/2]+C2e^[(-3-√5)t/2]+cost