Edo No Homogenea.docx

𝑴é𝒕𝒐𝒅𝒐 𝒂𝒃𝒓𝒆𝒗𝒊𝒂𝒅𝒐 𝒊𝒏𝒗𝒐𝒍𝒖𝒄𝒓𝒂𝒏𝒅𝒐 𝒐𝒑𝒆𝒓𝒂𝒅𝒐𝒓𝒆𝒔 𝒚′′ + 𝟐𝒚′ + 𝟐𝒚 = 𝟏 + 𝒙 𝐷 2 + 2𝐷 + 2 = 0 𝐷=

−𝑏 ± √𝑏 2 − 4𝑎𝑐 2𝑎

𝐷=

−2 ± √(2)2 − 4(1)(2) 2(1)

𝐷=

−2 ± √4 2

𝐷1 =

−2 + √−4 = −1 + 𝑖 2

𝐷2 =

−2 + √−4 = −1 − 𝑖 2

𝑅𝑎𝑖𝑐𝑒𝑠 𝑐𝑜𝑚𝑝𝑙𝑒𝑗𝑎𝑠 𝑐𝑜𝑛𝑗𝑢𝑔𝑎𝑑𝑎𝑠

𝒀𝒉 = 𝑪𝟏 𝒆−𝒙 𝐜𝐨𝐬 𝒙 + 𝑪𝟐 𝒆−𝒙 𝒔𝒆𝒏 𝒙 𝑆𝑖 𝑓(𝑥) = 𝑃(𝑥) 𝑓(𝑥) = 1 + 𝑥 𝑌𝑝 = 𝑃(𝑥) ∗ 𝑄(𝐷) 𝑄(𝐷) = 𝑌𝑝 = (1 + 𝑥) ∗

𝐷2

1 𝐹(𝐷)

1 + 2𝐷 + 2

1 −1 −𝐷 −𝐷 𝐷

1 − 𝐷2 2 1 − 𝐷2 2 +𝐷2 1 2 𝐷 2 1 − 𝐷2 2

2 +2𝐷 1 1 − 𝐷 2 2 1 + 𝐷3 2 1 + 𝐷3 2 1 + 𝐷3 2

1 1 1 𝑌𝑝 = (1 + 𝑥) ( − 𝐷 + 𝐷 2 ) 2 2 4 1 1 1 𝑌𝑝 = (1 + 𝑥) − 𝐷(1 + 𝑥) + 𝐷 2 (1 + 𝑥) 2 2 4

1 − 𝐷4 4 1 − 𝐷4 4

+𝐷2 1 + 𝐷2 4

𝑌𝑝 =

1 𝑥 1 𝑑 1 𝑑 (1 + 𝑥) + ∗ (1 + 𝑥) + − ∗ 2 2 2 𝑑𝑥 4 𝑑𝑥

𝑌𝑝 =

1 𝑥 1 1 𝑑 (1) + − ∗ (1) + ∗ 2 2 2 4 𝑑𝑥

𝑌𝑝 =

1 𝑥 1 1 + − ∗ (1) + ∗ (0) 2 2 2 4

1 𝑥 1 + − 2 2 2 𝒙 𝒀𝒑 = 𝟐 𝑌𝑝 =

𝑌 = 𝑌ℎ + 𝑌𝑝 𝑌 = 𝐶1 𝑒 −𝑥 cos 𝑥 + 𝐶2 𝑒 −𝑥 𝑠𝑒𝑛 𝑥 +

𝑥 2

𝒀 = 𝒆−𝒙 (𝑪𝟏 𝐜𝐨𝐬 𝒙 + 𝑪𝟐 𝒔𝒆𝒏 𝒙) +

𝒙 𝟐