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FastomWisp
@FastomWisp
August 2022
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9y''+6y'+y=0
y(0)=-5;y'(0)=4
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Verified answer
9y''+6y'+y=0
y(0)=-5;y'(0)=4
9k²+6k+1=0
(3k+1)²=0 ⇔ k1=k2=-1/3
У₁=e^(-x/3) У₂=x·e^(-x/3)
Y₀₀=C₁Y₁+C₂Y₂ Y₀₀=C₁e^(-x/3)+C₂x·e^(-x/3)=e^(-x/3)(C₁+C₂x)
Y'₀₀=[e^(-x/3)]'(C₁+C₂x)+[e^(-x/3)](C₁+C₂x)'=
= -([e^(-x/3)]/3 ·(C₁+C₂x)+[e^(-x/3)]C₂
y(0)=-5;y'(0)=4
Y₀₀(0)=C₁= -5 C₁= -5
Y'₀₀(0)=-(1/3) ·(C₁)+C₂=4 C₂=4+(-5/3)=7/3
Y₀₀=-5e^(-x/3)+(7/3)x·e^(-x/3)=e^(-x/3)(-5+7x/3)
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Answers & Comments
Verified answer
9y''+6y'+y=0y(0)=-5;y'(0)=4
9k²+6k+1=0
(3k+1)²=0 ⇔ k1=k2=-1/3
У₁=e^(-x/3) У₂=x·e^(-x/3)
Y₀₀=C₁Y₁+C₂Y₂ Y₀₀=C₁e^(-x/3)+C₂x·e^(-x/3)=e^(-x/3)(C₁+C₂x)
Y'₀₀=[e^(-x/3)]'(C₁+C₂x)+[e^(-x/3)](C₁+C₂x)'=
= -([e^(-x/3)]/3 ·(C₁+C₂x)+[e^(-x/3)]C₂
y(0)=-5;y'(0)=4
Y₀₀(0)=C₁= -5 C₁= -5
Y'₀₀(0)=-(1/3) ·(C₁)+C₂=4 C₂=4+(-5/3)=7/3
Y₀₀=-5e^(-x/3)+(7/3)x·e^(-x/3)=e^(-x/3)(-5+7x/3)