Factor Integrante.
𝑷(𝒙𝒚)𝑑𝑥 + 𝑸(𝒙𝒚)𝑑𝑦 = 0
𝜕𝑃(𝑥𝑦) 𝜕𝑄(𝑥𝑦)
𝑑𝑒𝑟𝑖𝑣𝑎𝑟
≠
𝑛𝑜 𝑖𝑔𝑢𝑎𝑙.
𝜕𝑦
𝜕𝑥
𝑢𝑠𝑎𝑟 𝐹𝑎𝑐𝑡𝑜𝑟 𝑖𝑛𝑡𝑒𝑔𝑟𝑎𝑛𝑡𝑒
1 𝜕𝑃 𝜕𝑄
( − ) = 𝐹(𝑥); 𝑢 = 𝑢(𝑥)𝑡𝑒𝑟𝑚𝑖𝑛𝑜 𝑥
𝑄 𝜕𝑦 𝜕𝑥
1 𝜕𝑄 𝜕𝑃
(
− ) = 𝐹(𝑦); 𝑢 = 𝑢(𝑦)𝑡𝑒𝑟𝑚𝑖𝑛𝑜 𝑦
𝑃 𝜕𝑥 𝜕𝑦
𝑠𝑎𝑐𝑎𝑟 𝑓(𝑥) 𝑜 𝑓(𝑦)
𝑑𝑢
𝑠𝑎𝑐𝑎𝑟𝐹. 𝑖𝑛𝑡𝑒𝑔𝑟𝑎𝑙 ∫
= ∫ 𝑓(𝑥) → 𝑢 =. .
𝑢
𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑐𝑎𝑟 𝑙𝑜 𝑞𝑢𝑒 𝑠𝑎𝑙𝑒 𝑢 𝑒𝑛 𝑒𝑙 𝑒𝑗𝑒𝑟𝑐𝑖𝑐𝑖𝑜 𝑑𝑎𝑑𝑜.
𝜕𝑃(𝑥𝑦) 𝜕𝑄(𝑥𝑦)
𝑈𝑡𝑖𝑙𝑖𝑧𝑎𝑟 𝑦 𝑑𝑒𝑟𝑖𝑣𝑎𝑟
=
𝑠𝑒𝑟 𝑖𝑔𝑢𝑎𝑙.
𝜕𝑦
𝜕𝑥
𝑅𝑒𝑠𝑜𝑙𝑣𝑒𝑟 𝑖𝑔𝑢𝑎𝑙𝐸𝐷𝐸𝑐𝑎𝑐𝑡𝑎
EDOrdinaria-ED
Demostración normal
𝑦 = 𝑓(𝑥, 𝑦) 𝑎 𝑙𝑎 𝑦 ′′ + 𝑦
=0
𝑑𝑦
=𝑓
𝑑𝑥
ED1ORDEN ED variable
separadas
𝐲′ = 𝐟(𝐱) ∗ 𝐠(𝐲)
dy
Y′ =
= f(x) ∗ g(y)
dx
∫
1
dy = f(x)dx
g(y)
𝐟(𝐱)𝐝𝐱 + 𝐠(𝐲)𝐝𝐲 = 𝟎
Demostrar paramétrica
𝑑𝑦
𝑥 = 𝑓(𝑡) ′ 𝑑𝑦
𝑦 =
= 𝑑𝑡
𝑦 = 𝑓(𝑡)
𝑑𝑥 𝑑𝑥
𝑑𝑦
∫ 𝑓𝑥dx + ∫ g(y)dy = 0
𝐟(𝐱)𝐠(𝐲)𝐝𝐱 + 𝐟𝟏(𝐱)𝐠𝟏(𝐲)𝐝𝐱
= 𝟎
𝑓(𝑥)
g1(y)
∫
dx + ∫
dy = 0
f1(x)
g(y)
ED Reducibles Variable Se.
y′ = f(𝐚𝐱 + 𝐛𝐲 + 𝐜)
𝑂𝑐𝑢𝑝𝑎 𝑠𝑢𝑠𝑡𝑖𝑡𝑢𝑐𝑖𝑜𝑛:
∗ u = ax + by + c
∗ du = adx + bdy
𝑑𝑦 𝑑𝑢
∗𝑏
=
−𝑎
𝑑𝑥 𝑑𝑥
𝑑𝑢 − 𝑎𝑑𝑥
∗ 𝑑𝑦 =
𝑏
EDHomogenea
𝐟(𝐭𝐱, 𝐭𝐲) = 𝒕𝒏 𝐟(𝐱, 𝐲) → ver si cumple
𝑬𝑫𝑯. 𝑒𝑛 𝑙𝑎 𝑞𝑢𝑒 𝑴 𝒚 𝑵 𝑠𝑜𝑛 𝑚𝑖𝑠𝑚𝑜 𝑮𝑹𝑨𝑫𝑶
𝑀(𝑥, 𝑦)𝑑𝑥 + 𝑁(𝑥, 𝑦)𝑑𝑦 = 0 → 𝐆𝐑𝐀𝐃𝐎 𝐈𝐆𝐔𝐀𝐋
Sustitución 1OrdenHomogenea
Sustitucion1y 2
𝑦
𝑑𝑦
𝑑𝑢
𝑢 = 𝑦 = 𝑢𝑥 𝑑𝑦 = 𝑢𝑑𝑥 + 𝑥𝑑𝑢
= 𝑢+𝑥
𝑥
𝑑𝑥
𝑑𝑥
𝑥
𝑑𝑥
𝑑𝑢
𝑢 = 𝑥 = 𝑢𝑦 𝑑𝑥 = 𝑢𝑑𝑦 + 𝑦𝑑𝑢
=𝑢+𝑦
𝑦
𝑑𝑦
𝑑𝑦
EDReducibleH.
a1x + b1y + c1
)
y=f (
a2x + b2y + c2
𝑎1 𝑏1
𝐴𝑝𝑙𝑖𝑐𝑎𝑟𝜆 = |
|=0𝑜 ≠0
𝑎2 𝑏2
𝑆𝑖𝑠𝑡𝑒𝑚𝑎 𝐸𝑐. ≠0
a1x + b1y + c1 = 0
{
a2x + b2y + c2 = 0
𝑥 = 𝑢 + 𝛼; 𝑑𝑥 = 𝑑𝑢
𝐴𝑝𝑙𝑖𝑐𝑜𝑠𝑢𝑠𝑡𝑖𝑡𝑢𝑐𝑖𝑜𝑛: {
𝑦 = 𝑣 + 𝛽; 𝑑𝑢 = 𝑑𝑦
𝑣
𝐿𝑢𝑒𝑔𝑜 𝑠𝑒 𝑎𝑝𝑙𝑖𝑐𝑎 EDVariSe 𝑡 = 𝑣 = 𝑡 ∗ 𝑢
𝑢
𝑑𝑣 = 𝑢𝑑𝑡 + 𝑡𝑑𝑢
𝑆𝑖𝑠𝑡𝑒𝑚𝑎 𝐸𝐶. =0
𝐴𝑝𝑙𝑖𝑐𝑜𝑆𝑢𝑠𝑡𝑖𝑡𝑢𝑐𝑖𝑜𝑛 "𝐸𝐷𝑉𝑎𝑟𝑖𝑆𝑒"
EDExacta.
𝐷𝑒𝑓. 𝑧 = 𝑓(𝑥, 𝑦) = 𝑑𝑧 = 𝑓𝑥𝑑𝑥 + 𝑓𝑦𝑑𝑦 𝐷𝑒𝑟. 𝑃𝑎𝑟𝑐𝑖.
𝐷𝑒𝑓. 𝐸𝐷𝐸𝑥𝑎𝑐𝑡𝑎 . 𝑷(𝒙𝒚)𝑑𝑥 + 𝑸(𝒙𝒚)𝑑𝑦 = 0
𝜕𝑃(𝑥𝑦) 𝜕𝑄(𝑥𝑦)
𝑑𝑒𝑟𝑖𝑣𝑎𝑟
=
𝑠𝑒𝑟 𝑖𝑔𝑢𝑎𝑙.
𝜕𝑦
𝜕𝑥
𝑟𝑒𝑒𝑚𝑝𝑙𝑎𝑧𝑎𝑟: 𝑢 = ∫ 𝑷(𝒙𝒚)𝑑𝑥 + φ(y)
𝑖𝑛𝑡𝑒𝑔𝑟𝑎𝑟 𝑟𝑒𝑠𝑝𝑒𝑐𝑡𝑜 X: 𝒖 = ⋯ + φ(y)
𝜕𝑢
𝐷𝑒𝑟𝑖𝑣𝑎𝑟 𝑟𝑒𝑠𝑝𝑒𝑐𝑡𝑜 𝑦:
=
𝜕𝑦
𝜕𝑢
= 𝑸(𝒙𝒚), 𝑑𝑒𝑠𝑝𝑒𝑗𝑎𝑟φ(y)
𝜕𝑦
𝑖𝑛𝑡𝑒𝑔𝑟𝑎𝑟φ(y) → ⋯ + 𝑐
𝑟𝑒𝑒𝑚𝑝𝑙𝑎𝑧𝑎𝑟 φ(y) 𝑒𝑛 𝒖. 𝐹𝐼𝑁
𝑖𝑔𝑢𝑎𝑙𝑎𝑟
EDLineales. 𝒂) 𝒚′ + 𝑷(𝒙) 𝑦 = 𝑸(𝒙)
𝑏) 𝑦 = 𝑐𝑒 − ∫ 𝑷(𝒙)𝒅𝒙
𝑅𝑒𝑠𝑜𝑙𝑣𝑒𝑟. 𝑑𝑒𝑠𝑝𝑒𝑗𝑎𝑟 𝑦 𝑇𝑖𝑒𝑛𝑒 𝑞𝑢𝑒 𝑡𝑒𝑛𝑒𝑟
𝑓𝑜𝑟𝑚𝑎: 𝒚′ + 𝑷(𝒙) 𝑦 = 𝑸(𝒙) 𝑜
𝒙′ + 𝑸(𝒚) 𝑥 = 𝑷(𝒚)
𝑙𝑢𝑒𝑔𝑜 𝑟𝑒𝑒𝑚𝑝𝑙𝑎𝑧𝑎𝑟 𝑒𝑛 𝒃) 𝒚 = 𝑪′ … (𝟏)
𝑙𝑢𝑒𝑔𝑜 𝑑𝑒𝑟𝑖𝑣𝑎𝑟 𝑦 ′ = 𝐶 ′
𝑙𝑢𝑒𝑔𝑜 𝑟𝑒𝑒𝑚𝑝𝑙𝑧𝑎 . 𝑒𝑛 𝒚′ + 𝑷(𝒙) 𝑦 = 𝑸(𝒙)
𝑠𝑒 𝑠𝑖𝑚𝑝𝑙𝑖𝑓𝑖𝑐𝑎𝑟𝑎 𝑙𝑢𝑒𝑔𝑜, 𝑑𝑒𝑠𝑝𝑒𝑗𝑎𝑟 𝑐(𝑥) ′ =. .
𝑝𝑜𝑟 𝑢𝑙𝑡𝑖𝑚𝑜 𝐼𝑛𝑡𝑒𝑔𝑟𝑎𝑟. 𝑦 𝑟𝑒𝑒𝑚𝑝𝑙𝑧𝑎𝑟 (𝟏)𝑭𝑰𝑵
EDLineales(MetodoSustitucion)
𝒚′ + 𝑷(𝒙) 𝑦 = 𝑸(𝒙)
𝑦 = 𝑢𝑣
𝑦 ′ = 𝑢𝑣 ′ + 𝑢′ 𝑣
𝑢𝑣 ′ + 𝑢′ 𝑣 + 𝑷(𝒙) 𝑢𝑣 = 𝑸(𝒙)
𝑣(𝒖′ + 𝑷(𝒙) 𝒗) + 𝑢𝑣 ′ = 𝑄(𝑥)
𝑖𝑛𝑡𝑒𝑔𝑟𝑎𝑟: 𝒖′ + 𝑷(𝒙) 𝒗
𝐼𝑛𝑡𝑒𝑔𝑟𝑎𝑟: 𝑢𝑣 ′ = 𝑄(𝑥)
𝑟𝑒𝑒𝑚𝑝𝑙𝑧𝑎𝑟 𝑦 = 𝑢𝑣