Variación de Parámetros Definición Ejercicio 1 𝑦´´ + 𝑦 = sec 2 𝑥 EHR 𝑦´´ + 𝑦 = 0 𝑦 = 𝑒 𝑚𝑥 𝑚2 + 1 = 0 𝑚 = 𝑎 ± 𝑏𝑖 𝑚2 = −1 𝑦 = 𝑒 𝑎𝑥 (𝐶1 cos 𝑏𝑥 + 𝐶2 sin 𝑏𝑥) 𝑚 = ± −1 𝑚 =0±𝑖 𝑚 = ±𝑖 𝑦 = 𝑒 0𝑥 (𝐶1 cos 𝑥 + 𝐶2 sin 𝑥) 𝑚= 𝑖 𝑦𝑐 = 𝐶1 cos 𝑥 + 𝐶2 sin 𝑥 𝑦𝑝 = 𝑢1 cos 𝑥 + 𝑢2 sin 𝑥 𝑢1 = − 𝑦2 𝑔(𝑥) 𝑑𝑥 𝑊 𝑦1 = cos 𝑥 𝑊= cos 𝑥 −sin 𝑥 𝑢2 = 𝑦2 = sin 𝑥 sin 𝑥 cos 𝑥 = cos 𝑥 cos 𝑥 − sin 𝑥 (−sin 𝑥) = cos 2 𝑥 + sin2 𝑥 =1 𝑦1 𝑔(𝑥) 𝑑𝑥 𝑊 𝑦1 𝑊 = 𝑦´ 1 𝑦2 𝑦´2 𝑔 𝑥 = sec 2 𝑥 𝑦1 = cos 𝑥 𝑦2 = sin 𝑥 𝑢1 = − 𝑦2 𝑔(𝑥) 𝑑𝑥 𝑊 𝑢1 = − sin 𝑥 sec 2 𝑥 𝑑𝑥 1 sec 𝑥 = 𝑢1 = − sin 𝑥 sec 𝑥 sec 𝑥 𝑑𝑥 𝑢1 = − 1 (sin 𝑥) (sec 𝑥) 𝑑𝑥 cos 𝑥 𝑢1 = − 𝑢1 = − 1 cos 𝑥 sin 𝑥 tan 𝑥 = cos 𝑥 sin 𝑥 (sec 𝑥) 𝑑𝑥 cos 𝑥 sec 𝑥 tan 𝑥 𝑑𝑥 𝑔 𝑥 = sec 2 𝑥 𝑢1 = − sec 𝑥 𝑊=1 𝑦1 = cos 𝑥 𝑢2 = 𝑦1 𝑔(𝑥) 𝑑𝑥 𝑊 𝑢2 = cos 𝑥 sec 2 𝑥 𝑑𝑥 1 𝑢2 = cos 𝑥 sec 2 𝑥 𝑑𝑥 𝑢2 = cos 𝑥 sec 𝑥 sec 𝑥 𝑑𝑥 𝑢2 = sec 𝑥 𝑑𝑥 𝑢2 = ln sec 𝑥 + tan 𝑥 𝑦2 = sin 𝑥 𝑔 𝑥 = sec 2 𝑥 𝑊=1 1 sec 𝑥 = → cos 𝑥 sec 𝑥 = 1 cos 𝑥 𝑦𝑝 = 𝑢1 cos 𝑥 + 𝑢2 sin 𝑥 𝑢1 = − sec 𝑥 𝑢2 = ln sec 𝑥 tan 𝑥 𝑦𝑝 = − sec 𝑥 cos 𝑥 + ln sec 𝑥 tan 𝑥 sin 𝑥 𝑦𝑝 = −1 + ln sec 𝑥 tan 𝑥 sin 𝑥 𝑦𝑝 = sin 𝑥 ln sec 𝑥 + tan 𝑥 − 1 𝑦 = 𝑦𝑐 + 𝑦𝑝 𝑦𝑐 = 𝐶1 cos 𝑥 + 𝐶2 sin 𝑥 𝑦 = 𝐶1 cos 𝑥 + 𝐶2 sin 𝑥 + sin 𝑥 ln sec 𝑥 +tan 𝑥 − 1 1 sec 𝑥 = cos 𝑥 Ejercicio 2 𝑦´´ − 𝑦 = 2 ln 𝑥 EHR 𝑦´´ − 𝑦 = 0 𝑦 = 𝑒 𝑚𝑥 𝑚2 − 1 = 0 𝑦1 = 𝑒 𝑥 (𝑚 − 1)(𝑚 + 1) = 0 𝑦2 = 𝑒 −𝑥 𝑚−1=0→ 𝑚 =1 𝑦𝑐 = 𝐶1 𝑒 𝑥 + 𝐶2 𝑒 −𝑥 𝑚 + 1 = 0 → 𝑚 = −1 𝑦𝑐 = 𝑢1 𝑒 𝑥 + 𝑢2 𝑒 −𝑥 𝑢1 = − 𝑦2 𝑔(𝑥) 𝑑𝑥 𝑊 𝑦1 = 𝑒 𝑥 𝑒𝑥 𝑊= 𝑥 𝑒 𝑢2 = 𝑦2 = 𝑒 −𝑥 𝑒 −𝑥 −𝑒 −𝑥 = 𝑒 𝑥 −𝑒 −𝑥 − (𝑒 −𝑥 )(𝑒 𝑥 ) = −𝑒 0 − 𝑒 0 = −1 − 1 = −2 𝑦1 𝑔(𝑥) 𝑑𝑥 𝑊 𝑦1 𝑊 = 𝑦´ 1 𝑦2 𝑦´2 𝑔 𝑥 = 2 ln 𝑥 𝑦1 = 𝑒 𝑥 𝑦2 = 𝑒 −𝑥 𝑢1 = − 𝑦2 𝑔(𝑥) 𝑑𝑥 𝑊 𝑢1 = − 𝑒 −𝑥 2 ln 𝑥 𝑑𝑥 −2 𝑢1 = 𝑒 −𝑥 ln 𝑥 𝑑𝑥 𝑔 𝑥 = 2 ln 𝑥 𝑢2 = 𝑦1 𝑔(𝑥) 𝑑𝑥 𝑊 𝑢2 = 𝑒 −𝑥 2 ln 𝑥 𝑑𝑥 −2 𝑢2 = − 𝑒 𝑥 ln 𝑥 𝑑𝑥 𝑊 = −2 𝑦𝑝 = 𝑢1 𝑒 𝑥 + 𝑢2 𝑒 −𝑥 𝑢1 = 𝑒 −𝑥 ln 𝑥 𝑑𝑥 𝑦𝑝 = 𝑒 𝑥 𝑢2 = − 𝑒 −𝑥 ln 𝑥 𝑑𝑥 − 𝑒 −𝑥 𝑦 = 𝑦𝑐 + 𝑦𝑝 𝑦 = 𝐶1 𝑒 𝑥 + 𝐶2 𝑒 −𝑥 + 𝑒 𝑥 𝑒 𝑥 ln 𝑥 𝑑𝑥 𝑒 𝑥 ln 𝑥 𝑑𝑥 𝑦𝑐 = 𝐶1 𝑒 𝑥 + 𝐶2 𝑒 −𝑥 𝑒 −𝑥 ln 𝑥 𝑑𝑥 − 𝑒 −𝑥 𝑒 𝑥 ln 𝑥 𝑑𝑥 Ejercicio 3 𝑒 2𝑥 𝑦´´ − 4𝑦´ + 4𝑦 = 2 𝑥 EHR 𝑦´´ − 4𝑦´ + 4𝑦 = 0 𝑦 = 𝑒 𝑚𝑥 𝑚2 − 4𝑚 + 4 = 0 𝑦1 = 𝑒 2𝑥 (𝑚 − 2)(𝑚 − 2) = 0 𝑦2 = 𝑥𝑒 2𝑥 𝑚−2=0 𝑚=2 𝑦𝑐 = 𝐶1 𝑒 2𝑥 + 𝐶2 𝑥𝑒 2𝑥 𝑦𝑝 = 𝑢1 𝑒 2𝑥 + 𝑢2 𝑥𝑒 2𝑥 𝑢1 = − 𝑦2 𝑔(𝑥) 𝑑𝑥 𝑊 𝑦1 = 𝑒 2𝑥 2𝑥 𝑒 𝑊= 2𝑒 2𝑥 𝑢2 = 𝑦1 𝑔(𝑥) 𝑑𝑥 𝑊 𝑦2 = 𝑥𝑒 2𝑥 𝑥𝑒 2𝑥 𝑒 2𝑥 + 2𝑥𝑒 2𝑥 = 𝑒 2𝑥 𝑒 2𝑥 + 2𝑥𝑒 2𝑥 − (𝑥𝑒 2𝑥 )(2𝑒 2𝑥 ) = 𝑒 4𝑥 𝑥 + 2𝑥𝑒 4𝑥 − 2𝑥𝑒 4𝑥 = 𝑒 4𝑥 𝑦1 𝑊 = 𝑦´ 1 𝑔 𝑥 = 𝑒 2𝑥 𝑥2 𝑦2 𝑦´2 𝑦1 = 𝑒 2𝑥 𝑢1 = − 𝑦2 = 𝑦2 𝑔(𝑥) 𝑑𝑥 𝑊 𝑥𝑒 2𝑥 𝑢1 = − 𝑢1 = − 𝑢1 = − 𝑒 2𝑥 𝑥 2 𝑑𝑥 𝑒 4𝑥 𝑥 𝑒 4𝑥 2 𝑥 𝑑𝑥 𝑒 4𝑥 1 𝑑𝑥 𝑥 𝑥𝑒 2𝑥 𝑔 𝑥 = 𝑢2 = 𝑒 2𝑥 𝑊 = 𝑒 4𝑥 𝑥2 𝑦1 𝑔(𝑥) 𝑑𝑥 𝑊 𝑒 2𝑥 𝑥 2 𝑑𝑥 4𝑥 𝑒 2𝑥 𝑢2 = 𝑒 𝑒 4𝑥 𝑢2 = 1 𝑥 2 𝑑𝑥 𝑒 4𝑥 𝑢2 = 1 𝑑𝑥 2 𝑥 𝑢2 = 𝑥 −2 𝑑𝑥 𝑢1 = − ln 𝑥 𝑥 −2+1 𝑢2 = −2 + 1 𝑛+1 𝑥 𝑥 𝑛 𝑑𝑥 = 𝑛+1 𝑥 −1 𝑢2 = −1 1 𝑢2 = − 𝑥 𝑦𝑝 = 𝑢1 𝑒 2𝑥 + 𝑢2 𝑥𝑒 2𝑥 𝑢1 = − ln 𝑥 1 𝑢2 = − 𝑥 1 2𝑥 𝑦𝑝 = − 𝑥𝑒 𝑥 𝑦𝑝 = − ln 𝑥 𝑒 2𝑥 − 𝑒 2𝑥 − ln 𝑥 𝑒 2𝑥 𝑦 = 𝑦𝑐 + 𝑦𝑝 𝑦 = 𝐶1 𝑒 2𝑥 + 𝐶2 𝑥𝑒 2𝑥 − ln 𝑥 𝑒 2𝑥 − 𝑒 2𝑥 𝑦 = 𝐶1 𝑒 2𝑥 + 𝐶2 𝑥𝑒 2𝑥 − ln 𝑥 𝑒 2𝑥 𝑦𝑐 = 𝐶1 𝑒 2𝑥 + 𝐶2 𝑥𝑒 2𝑥