Ecuaciones diferenciales reducibles a homogéneas.
Le llamaremos ecuación diferencial reducible a homogénea de primer grado a
𝑑𝑦 𝑎𝑥+𝑏𝑦+𝑐
la ecuación de la forma = 𝑓( ) la cual es posible reducir a homogénea
𝑑𝑥 𝑑𝑥+𝑒𝑦+𝑓
trasladando el punto de intersección entre las rectas (𝑥0 , 𝑦0 ) al origen de las
coordenadas, con la excepción de que si las rectas son paralelas el método no se cumple.
Así 𝑥 = ℎ + 𝑥0 , 𝑦 = 𝑘 + 𝑦0 .
Ejemplo 22. Resuelva la siguiente ecuación reducible a homogénea.
(𝑥 + 𝑦 − 2)𝑑𝑥 + (𝑥 − 𝑦 + 4)𝑑𝑦 = 0
Solución.
(𝑥 + 𝑦 − 2)𝑑𝑥 + (𝑥 − 𝑦 + 4)𝑑𝑦 = 0 primero encontramos la intersección entre las rectas.
𝑥 + 𝑦 − 2 = 0 …(1) y 𝑥 − 𝑦 + 4 = 0 …(2) si despejamos 𝑥 de la ecuación(2), 𝑥 = 𝑦 − 4 y lo sustituimos en la ecuación (1)
Problema 25. Resuelva la siguiente ecuación reducible a lineal de segundo grado.
𝑦 3 𝑑𝑥 + 2(𝑥 2 − 𝑥𝑦 3 )𝑑𝑦 = 0
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