常微分方程题目：求下组初值问题的解x''+9x=6e^(3t),x(0)=x'(0)=0

Question

常微分方程题目：求下组初值问题的解
x''+9x=6e^(3t) ,x(0)=x'(0)=0

Answer 1

λ^2+9=0
λ=±3i
设y*=ae^(3t)
9ae^(3t)+9ae^(3t)=6e^(3t) a=1/3
通解x=c1sin3t+c2cos3t+(1/3)e^(3t)
x'=3c1cos3t-3c2sin3t+e^(3t)
0=c2+(1/3) c2=-1/3
0=3c1+1 c1=-1/3
x=(-1/3)sin3t+(-1/3)cos3t+(1/3)e^(3t)