y′′+y=2x+3ex
1)y′′+y=0
k2+1=0
k1=i,k2=−i
y=C1cos(x)+C2sin(x)
k1+ik2=i+1=k1=k2
y=ek1An(x)=Aei
2)y′′+y=2x+3ex
y=Aex+Bx
y′=Aex+B
y′′=Aex
y′′+y′=Aex+Aex+Bx=2x+3ex
{2A=3=>A=3/2B=2
y=C1cos(x)+C2sin(x)+23ex+2x
Checking:
y′′=−C1cos(x)−C2sin(x)+23ex
y′′+y=−C1cos(x)−C2sin(x)+23ex+C1cos(x)+C2sin(x)+23ex+2x=3ex+2x
Answer:
y=C1cos(x)+C2sin(x)+23ex+2x
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