9y''+6y'+y=0

y (0) = - 5; y' (0) = 4

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)