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Asignacion de Calculo4 Carlos gonzalez Saia E

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UNIVERSIDAD FERMIN TORO VICERECTORADO ACADEMICO DECANATO DE INGENIERIA MATEMATICA IV ACTIVIDAD EJERCICIOS Carlos Gonzรกlez 24538470
1.) Determine si la funciรณn es soluciรณn de la ecuaciรณn diferencial. a y ๏€ฝ senx ๏€จ x ๏€ซ ctgx๏€ฉ y ๏€ซ y ๏€ฝ ๏€ญctgx ,, ) ln csc ; 1) ํ‘ฆ = ํ‘ ํ‘’ํ‘› ํ‘ฅln (csc ํ‘ฅ + ํ‘ํ‘œํ‘กํ‘” ํ‘ฅ) ํ‘ฆยด = ํ‘ํ‘œํ‘ ํ‘ฅ ln(csc ํ‘ฅ + ํ‘ํ‘œํ‘กํ‘” ํ‘ฅ) + 1 csc ํ‘ฅ + ํ‘ํ‘œํ‘กํ‘” ํ‘ฅ (โˆ’ csc ํ‘ฅํ‘ํ‘œํ‘กํ‘” ํ‘ฅ) โˆ’ csc 2 ํ‘ฅ ํ‘ ํ‘’ํ‘› ํ‘ฅ ํ‘ฆยด = ํ‘ํ‘œํ‘ ํ‘ฅ ln(csc ํ‘ฅ + ํ‘ํ‘œํ‘กํ‘” ํ‘ฅ) + 1 csc ํ‘ฅ + ํ‘ํ‘œํ‘กํ‘” ํ‘ฅ (โˆ’ csc ํ‘ฅ)(ํ‘ํ‘œํ‘กํ‘” ํ‘ฅ +csc ํ‘ฅ)ํ‘ ํ‘’ํ‘› ํ‘ฅ ํ‘ฆยด = ํ‘ํ‘œํ‘ ํ‘ฅ ln(csc ํ‘ฅ + ํ‘ํ‘œํ‘กํ‘” ํ‘ฅ) โˆ’ csc ํ‘ฅ ํ‘ ํ‘’ํ‘› ํ‘ฅ ํ‘ฆยด = ํ‘ํ‘œํ‘ ํ‘ฅ ln(csc ํ‘ฅ + ํ‘ํ‘œํ‘กํ‘” ํ‘ฅ) โˆ’ 1 ํ‘ฆยดยด = โˆ’ํ‘ ํ‘’ํ‘› ํ‘ฅ ln(csc ํ‘ฅ + ํ‘ํ‘œํ‘กํ‘” ํ‘ฅ) + 1 csc ํ‘ฅ + ํ‘ํ‘œํ‘กํ‘” ํ‘ฅ (โˆ’ csc ํ‘ฅํ‘ํ‘œํ‘กํ‘” ํ‘ฅ โˆ’ csc 2 ํ‘ฅ) ํ‘ํ‘œํ‘  ํ‘ฅ ํ‘ฆยดยด = โˆ’ํ‘ ํ‘’ํ‘› ํ‘ฅ ln(csc ํ‘ฅ + ํ‘ํ‘œํ‘กํ‘” ํ‘ฅ) + 1 csc ํ‘ฅ + ํ‘ํ‘œํ‘กํ‘” ํ‘ฅ (โˆ’ csc ํ‘ฅ)(ํ‘ํ‘œํ‘กํ‘” ํ‘ฅ +csc ํ‘ฅ)ํ‘ํ‘œํ‘  ํ‘ฅ ํ‘ฆยดยด = โˆ’ํ‘ ํ‘’ํ‘› ํ‘ฅ ln(csc ํ‘ฅ + ํ‘ํ‘œํ‘กํ‘” ํ‘ฅ) โˆ’ csc ํ‘ฅ cos ํ‘ฅ ํ‘ฆยดยด = โˆ’ํ‘ ํ‘’ํ‘› ํ‘ฅ ln(csc ํ‘ฅ + ํ‘ํ‘œํ‘กํ‘” ํ‘ฅ) โˆ’ 1 ํ‘ ํ‘’ํ‘›ํ‘ฅ ํ‘ํ‘œํ‘  ํ‘ฅ ํ‘ฆยดยด = โˆ’ํ‘ ํ‘’ํ‘› ํ‘ฅ ln(csc ํ‘ฅ + ํ‘ํ‘œํ‘กํ‘” ํ‘ฅ) โˆ’ ํ‘ํ‘œํ‘กํ‘”ํ‘ฅ ํ‘ฆยดยด + ํ‘ฆ = โˆ’ํ‘ํ‘œํ‘กํ‘” ํ‘ฅ โˆ’ํ‘ ํ‘’ํ‘› ํ‘ฅ ln(csc ํ‘ฅ + ํ‘ํ‘œํ‘กํ‘” ํ‘ฅ) โˆ’ ํ‘ํ‘œํ‘กํ‘”ํ‘ฅ + ํ‘ ํ‘’ํ‘› ํ‘ฅ ln(csc ํ‘ฅ + ํ‘ํ‘œํ‘กํ‘” ํ‘ฅ) = โˆ’ํ‘ํ‘œํ‘กํ‘” ํ‘ฅ โˆ’ํ‘ํ‘œํ‘กํ‘” ํ‘ฅ = โˆ’ํ‘ํ‘œํ‘กํ‘”ํ‘ฅ
2) Resolver las siguientes ecuaciones diferenciales de primer orden de acuerdo al mรฉtodo correspondiente. a .) senx cos xy y tg x , 2 ๏€ซ ๏€ฝ ) ๏€จ 2 y cos ๏€ฉ ๏€จ2 2 y cos 2 ๏€ฉ 0 b e y xy dx xe x xy y dy ๏€ญ ๏€ซ ๏€ญ ๏€ซ ๏€ฝ 2- a) ํ‘ ํ‘’ํ‘›ํ‘ฅ cos ํ‘ฅ ํ‘‘ํ‘ฆ ํ‘‘ํ‘ฅ + ํ‘ฆ = ํ‘กํ‘”2ํ‘ฅ ํ‘ ํ‘’ํ‘›ํ‘ฅ cos ํ‘ฅ ํ‘‘ํ‘ฆ ํ‘‘ํ‘ฅ = (ํ‘กํ‘”2ํ‘ฅ โˆ’ ํ‘ฆ) ํ‘ ํ‘’ํ‘›ํ‘ฅ cos ํ‘ฅํ‘‘ํ‘ฆ = (ํ‘กํ‘”2ํ‘ฅ โˆ’ ํ‘ฆ)ํ‘‘ํ‘ฅ ํ‘ ํ‘’ํ‘›ํ‘ฅ cos ํ‘ฅ ํ‘‘ํ‘ฆ + (ํ‘ฆโˆ’ํ‘กํ‘”2ํ‘ฅ)ํ‘‘ํ‘ฅ = 0 ํ‘€ = ํ‘ฆโˆ’ํ‘กํ‘”2ํ‘ฅ ํ‘€ํ‘ฆ = 1 ํ‘ = ํ‘ ํ‘’ํ‘›ํ‘ฅ cos ํ‘ฅ ํ‘ํ‘ฅ = ํ‘ํ‘œํ‘ ํ‘ฅ โˆ— ํ‘ํ‘œํ‘ ํ‘ฅ + ํ‘ ํ‘’ํ‘›ํ‘ฅ โˆ— (โˆ’ํ‘ ํ‘’ํ‘›ํ‘ฅ) = ํ‘ํ‘œํ‘ 2ํ‘ฅ โˆ’ ํ‘ ํ‘’ํ‘›2ํ‘ฅ ํ‘€ํ‘ฆ โˆ’ ํ‘ํ‘ฅ ํ‘ = 1 โˆ’ ํ‘ํ‘œํ‘ 2ํ‘ฅ + ํ‘ ํ‘’ํ‘›2ํ‘ฅ ํ‘ ํ‘’ํ‘› ํ‘ฅ cos ํ‘ฅ = 2ํ‘ ํ‘’ํ‘›2ํ‘ฅ ํ‘ ํ‘’ํ‘› ํ‘ฅ cos ํ‘ฅ = 2ํ‘ ํ‘’ํ‘›ํ‘ฅ cos ํ‘ฅ = 2ํ‘กํ‘”ํ‘ฅ ํ‘’ โˆซ 2ํ‘กํ‘”ํ‘ฅํ‘‘ํ‘ฅ = ํ‘’ln(ํ‘ ํ‘’ํ‘ํ‘ฅ)2 = (ํ‘ ํ‘’ํ‘ํ‘ฅ)2 (ํ‘ ํ‘’ํ‘ํ‘ฅ)2ํ‘ ํ‘’ํ‘›ํ‘ฅcos ํ‘ฅ ํ‘‘ํ‘ฆ + (ํ‘ ํ‘’ํ‘ํ‘ฅ)2(ํ‘ฆโˆ’ํ‘กํ‘”2ํ‘ฅ)ํ‘‘ํ‘ฅ = 0 ํ‘กํ‘”ํ‘ฅ ํ‘‘ํ‘ฆ + (ํ‘ฆํ‘ ํ‘’ํ‘2ํ‘ฅ โˆ’ ํ‘ ํ‘’ํ‘2ํ‘ฅํ‘กํ‘”2ํ‘ฅ)ํ‘‘ํ‘ฅ = 0 ํ‘กํ‘”ํ‘ฅํ‘‘ํ‘ฆ + (ํ‘ฆํ‘ ํ‘’ํ‘2ํ‘ฅ โˆ’ ํ‘ ํ‘’ํ‘2ํ‘ฅํ‘กํ‘”2ํ‘ฅ)ํ‘‘ํ‘ฅ = 0 ํ‘€ = ํ‘ฆํ‘ ํ‘’ํ‘2ํ‘ฅ โˆ’ ํ‘ ํ‘’ํ‘2ํ‘ฅํ‘กํ‘”2ํ‘ฅ ํ‘€ํ‘ฆ = ํ‘ ํ‘’ํ‘2ํ‘ฅ ํ‘ = ํ‘กํ‘”ํ‘ฅ ํ‘ํ‘ฅ = sec 2 ํ‘ฅ
โˆซ ํ‘‘ํ‘“ = โˆซ ํ‘€ํ‘‘ํ‘ฅ โˆซ ํ‘‘ํ‘“ = โˆซ(ํ‘ฆํ‘ ํ‘’ํ‘2ํ‘ฅ โˆ’ ํ‘ ํ‘’ํ‘2ํ‘ฅํ‘กํ‘”2ํ‘ฅ)ํ‘‘ํ‘ฅ ํ‘“ = ํ‘ฆํ‘กํ‘”ํ‘ฅ โˆ’ ํ‘กํ‘”3 3 + โ„Ž(ํ‘ฆ) ํ‘“ํ‘ฆ = ํ‘กํ‘”ํ‘ฅ โˆ’ โ„Žโ€ฒ(ํ‘ฆ) ํ‘“ํ‘ฆ = ํ‘ ํ‘กํ‘”ํ‘ฅ + โ„Žโ€ฒ(ํ‘ฆ) = ํ‘กํ‘”ํ‘ฅ โˆซ ํ‘‘โ„Ž(ํ‘ฆ) = โˆซ 0ํ‘‘ํ‘ฆ โ„Ž(ํ‘ฆ) = ํถ ํ‘“ = ํ‘ฆํ‘กํ‘”ํ‘ฅ + ํ‘กํ‘”3ํ‘ฅ 3 + ํ‘ 2.b-) (ํ‘’2ํ‘ฆ โˆ’ ํ‘ฆํ‘ํ‘œํ‘  ํ‘ฅํ‘ฆ)ํ‘‘ํ‘ฅ + (2ํ‘ฅํ‘’2ํ‘ฆ โˆ’ ํ‘ฅํ‘ํ‘œํ‘  ํ‘ฅํ‘ฆ + 2ํ‘ฆ)ํ‘‘ํ‘ฆ = 0 ํ‘€(ํ‘ฅ, ํ‘ฆ) = ํ‘’2ํ‘ฆ โˆ’ ํ‘ฆํ‘ํ‘œํ‘  ํ‘ฅํ‘ฆ ํ‘(ํ‘ฅ, ํ‘ฆ) = 2ํ‘ฅํ‘’2ํ‘ฆ โˆ’ ํ‘ฅํ‘ํ‘œํ‘  ํ‘ฅํ‘ฆ + 2ํ‘ฆ ํ‘€ํ‘ฆ = 2ํ‘’2ํ‘ฆ โˆ’ 1cos ํ‘ฅํ‘ฆ + (โˆ’ํ‘ฆ)(โˆ’ํ‘ ํ‘’ํ‘› ํ‘ฅํ‘ฆ)ํ‘ฅ = 2ํ‘’2ํ‘ฆ โˆ’ cos ํ‘ฅํ‘ฆ + (ํ‘ฅํ‘ฆํ‘ ํ‘’ํ‘› ํ‘ฅํ‘ฆ) ํ‘ํ‘ฅ = 2ํ‘’2ํ‘ฆ โˆ’ 1 cos ํ‘ฅํ‘ฆ + (โˆ’ํ‘ฅ)(โˆ’ํ‘ ํ‘’ํ‘› ํ‘ฅํ‘ฆ)ํ‘ฆ = 2ํ‘’2ํ‘ฆ โˆ’ cos ํ‘ฅํ‘ฆ + (ํ‘ฅํ‘ฆํ‘ ํ‘’ํ‘› ํ‘ฅํ‘ฆ) ํ‘€ํ‘ฆ = ํ‘ํ‘ฅ ํธํ‘ฅํ‘Žํ‘ํ‘กํ‘Ž ํœ•ํ‘“(ํ‘ฅ, ํ‘ฆ) ํœ•ํ‘ฅ = ํ‘€(ํ‘ฅ, ํ‘ฆ) โˆซ ํœ•ํ‘“(ํ‘ฅ, ํ‘ฆ) = โˆซ(ํ‘’2ํ‘ฆ โˆ’ ํ‘ฆํ‘ํ‘œํ‘  ํ‘ฅํ‘ฆ)ํ‘‘ํ‘ฅ = ํ‘“(ํ‘ฅ, ํ‘ฆ) = ํ‘ฅํ‘’2ํ‘ฆ โˆ’ ํ‘ฆํ‘ ํ‘’ํ‘› ํ‘ฅํ‘ฆ ํ‘ฆ + โ„Ž(ํ‘ฆ) ํ‘“(ํ‘ฅ, ํ‘ฆ) = ํ‘ฅํ‘’2ํ‘ฆ โˆ’ ํ‘ ํ‘’ํ‘› ํ‘ฅํ‘ฆ + โ„Ž(ํ‘ฆ)
ํœ•ํ‘“(ํ‘ฅ, ํ‘ฆ) ํœ•ํ‘ฆ = 2ํ‘ฅํ‘’2ํ‘ฆ โˆ’ ํ‘ฅํ‘ํ‘œํ‘  ํ‘ฅํ‘ฆ + โ„Žยด(ํ‘ฆ) ํœ•ํ‘“(ํ‘ฅ, ํ‘ฆ) ํœ•ํ‘ฆ = ํ‘(ํ‘ฅ, ํ‘ฆ) 2ํ‘ฅํ‘’2ํ‘ฆ โˆ’ ํ‘ฅํ‘ํ‘œํ‘  ํ‘ฅํ‘ฆ + โ„Žยด(ํ‘ฆ) = 2ํ‘ฅํ‘’2ํ‘ฆ โˆ’ ํ‘ฅํ‘ํ‘œํ‘  ํ‘ฅํ‘ฆ + 2ํ‘ฆ โ„Žยด(ํ‘ฆ) = 2ํ‘ฆ 2 โˆซ โ„Ž(ํ‘ฆ)ํ‘‘ํ‘ฆ โ„Ž(ํ‘ฆ) = ํ‘ฆ2 + ํถ ํ‘“(ํ‘ฅ, ํ‘ฆ) = ํ‘ฅํ‘’2ํ‘ฆ โˆ’ ํ‘ ํ‘’ํ‘› ํ‘ฅํ‘ฆ + ํ‘ฆ2 + ํถ 3.) Resolver las ecuaciones diferenciales de orden N por coeficientes indeterminados. a .) y ,, ๏€ซ y , ๏€ฝ 2 e 2 x senx .) ,, 9 93 3cos b y ๏€ซ y ๏€ฝ x ๏€ซ x 3 a) ํ‘ฆยดยด + ํ‘ฆยด = 2ํ‘’2ํ‘ฅ ํ‘ ํ‘’ํ‘› ํ‘ฅ Para la soluciรณn homogenea ํ‘š2 + ํ‘š = 0 ํ‘š = 0 ํ‘œ ํ‘š = โˆ’1 ํ‘ฆโ„Ž = ํ‘1 + ํ‘2ํ‘’โˆ’ํ‘ฅ ํท(ํท + 1)(ํท2 โˆ’ 4ํท + 5)ํ‘ฆ = 0 ํท(ํท + 1)(ํท โˆ’ 2 โˆ’ ํ‘–)(ํท โˆ’ 2 + ํ‘–)ํ‘ฆ = 0 ํ‘ฆ = ํ‘1 + ํ‘2ํ‘’โˆ’ํ‘ฅ + ํดํ‘’2ํ‘ฅ ํ‘ํ‘œํ‘ ํ‘ฅ + ํตํ‘’2ํ‘ฅ ํ‘ ํ‘’ํ‘›ํ‘ฅ
Yh yp ํ‘ฆยดํ‘ = ํด2ํ‘’2ํ‘ฅ ํ‘ํ‘œํ‘ ํ‘ฅ + (โˆ’ํ‘ ํ‘’ํ‘›ํ‘ฅ)ํดํ‘’2ํ‘ฅ + ํต2ํ‘’2ํ‘ฅ ํ‘ ํ‘’ํ‘›ํ‘ฅ + ํ‘ํ‘œํ‘ ํ‘ฅํตํ‘’2ํ‘ฅ = (2ํด + ํต)ํ‘’2ํ‘ฅ ํ‘ํ‘œํ‘ ํ‘ฅ + (โˆ’ํด + 2ํต)ํ‘’2ํ‘ฅ ํ‘ ํ‘’ํ‘›ํ‘ฅ ํ‘ฆยดยดํ‘ = 2(2ํด + ํต)ํ‘’2ํ‘ฅ ํ‘ํ‘œํ‘ ํ‘ฅ + (2ํด + ํต)ํ‘’2ํ‘ฅ (โˆ’ํ‘ ํ‘’ํ‘›ํ‘ฅ) + 2(โˆ’ํด + 2ํต)ํ‘’2ํ‘ฅ ํ‘ ํ‘’ํ‘›ํ‘ฅ + (โˆ’ํด + 2ํต)ํ‘’2ํ‘ฅ ํ‘ํ‘œํ‘ ํ‘ฅ = (3ํด + ํต)ํ‘’2ํ‘ฅ ํ‘ํ‘œํ‘ ํ‘ฅ + (โˆ’4ํด + 3ํต)ํ‘’2ํ‘ฅ ํ‘ ํ‘’ํ‘›ํ‘ฅ ํ‘ฆยดยดํ‘ + ํ‘ฆยดํ‘ = (5ํด + 5ํต)ํ‘’2ํ‘ฅ ํ‘ํ‘œํ‘ ํ‘ฅ + (โˆ’5ํด + 5ํต)ํ‘’2ํ‘ฅ ํ‘ ํ‘’ํ‘›ํ‘ฅ = 2ํ‘’2ํ‘ฅ ํ‘ ํ‘’ํ‘›ํ‘ฅ โˆ’5ํด + 5ํต = 2 5ํด + 5ํต = 0 ํด = โˆ’ 1 5 ํ‘ฆ ํต = 1 5 ํ‘ฆ = ํ‘1 + ํ‘2ํ‘’โˆ’ํ‘ฅ โˆ’ 1 5 ํ‘’2ํ‘ฅ ํ‘ํ‘œํ‘ ํ‘ฅ + 1 5 ํ‘’2ํ‘ฅ ํ‘ ํ‘’ํ‘›ํ‘ฅ 3 b ) ํ‘ฆยดยด + 9ํ‘ฆ = 93ํ‘ฅ + 3ํ‘ํ‘œํ‘ ํ‘ฅ (ํท2 + 9)ํท2(ํท2 + 1)ํ‘ฆ = 0
ํ‘ฆ = ํถ1ํ‘ํ‘œํ‘ 3ํ‘ฅ + 2ํถํ‘ ํ‘’ํ‘›3ํ‘ฅ + ํถ3 + ํถ4ํ‘ฅ + ํถ5ํ‘ํ‘œํ‘ ํ‘ฅ + ํถ6ํ‘ ํ‘’ํ‘›ํ‘ฅ Yh yp ํ‘ฆยดํ‘ = ํถ4 โˆ’ ํถ5ํ‘ ํ‘’ํ‘›ํ‘ฅ + ํถ6ํ‘ํ‘œํ‘ ํ‘ฅ ํ‘ฆยดยดํ‘ = โˆ’ํถ5ํ‘ํ‘œํ‘ ํ‘ฅ โˆ’ ํถ6ํ‘ ํ‘’ํ‘›ํ‘ฅ ํ‘ฆยดยดํ‘ + 9ํ‘ฆํ‘ = 93ํ‘ฅ + 3ํ‘ํ‘œํ‘ ํ‘ฅ โˆ’ํถ5ํ‘ํ‘œํ‘ ํ‘ฅ โˆ’ ํถ6ํ‘ ํ‘’ํ‘›ํ‘ฅ + 9(ํถ3 + ํถ4ํ‘ฅ + ํถ5ํ‘ํ‘œํ‘ ํ‘ฅ + ํถ6ํ‘ ํ‘’ํ‘›ํ‘ฅ) = 93ํ‘ฅ + 3ํ‘ํ‘œํ‘ ํ‘ฅ 9ํถ3 + 9ํถ4ํ‘ฅ + 8ํถ5ํ‘ํ‘œํ‘ ํ‘ฅ + 8ํถ6ํ‘ ํ‘’ํ‘›ํ‘ฅ = 93ํ‘ฅ + 3ํ‘ํ‘œํ‘ ํ‘ฅ 9ํถ3 = 0 ํถ3 = 0 9ํถ4 = 93 ํถ4 = 93 9 8ํถ5 = 3 ํถ5 = 3 8 8ํถ6 = 0 ํถ6 = 0 ํ‘ฆ = ํถ1ํ‘ํ‘œํ‘ 3ํ‘ฅ + 2ํถํ‘ ํ‘’ํ‘›3ํ‘ฅ + 0 + ํถ4ํ‘ฅ + ํถ5ํ‘ํ‘œํ‘ ํ‘ฅ + 0ํ‘ ํ‘’ํ‘›ํ‘ฅ ํ‘ฆ = ํถ1ํ‘ํ‘œํ‘ 3ํ‘ฅ + ํถ2ํ‘ ํ‘’ํ‘›3ํ‘ฅ + 93 9 ํ‘ฅ + 3 8 ํ‘ํ‘œํ‘ ํ‘ฅ
4.) Resuelva por variaciรณn de parรกmetros: 1 9 ,, ๏€ซ ๏€ฝ 4 y y Csc3x 4 a) ํ‘ฆยดยด + 9ํ‘ฆ = 1 4 csc 3ํ‘ฅ ํ‘š2 + 9 = 0 ํ‘ฆโ„Ž = ํ‘1ํ‘ํ‘œํ‘ 3ํ‘ฅ + ํ‘2ํ‘ ํ‘’ํ‘›3ํ‘ฅ y1 y2 ํ‘ฆ1โ€ฒ = โˆ’3ํ‘ ํ‘’ํ‘›3ํ‘ฅ ํ‘ฆ2โ€ฒ = 3ํ‘ํ‘œํ‘ 3ํ‘ฅ ํ‘“(ํ‘ฅ) = 1 4 csc 3ํ‘ฅ ํ‘ค = | ํ‘ํ‘œํ‘ 3ํ‘ฅ ํ‘ ํ‘’ํ‘›3ํ‘ฅ โˆ’3ํ‘ ํ‘’ํ‘›3ํ‘ฅ 3ํ‘ํ‘œํ‘ 3ํ‘ฅ | = 3ํ‘ํ‘œํ‘ 23ํ‘ฅ + 3ํ‘ ํ‘’ํ‘›23ํ‘ฅ = 3 ํ‘ค1 = | 0 ํ‘ ํ‘’ํ‘›3ํ‘ฅ 1 4 csc 3ํ‘ฅ 3ํ‘ํ‘œํ‘ 3ํ‘ฅ | = 0 โˆ’ 1 4 csc 3ํ‘ฅํ‘ ํ‘’ํ‘›3ํ‘ฅ = โˆ’ 1 4 1 ํ‘ ํ‘’ํ‘›3ํ‘ฅ ํ‘ ํ‘’ํ‘›3ํ‘ฅ = โˆ’ 1 4 ํ‘ค2 = | ํ‘ํ‘œํ‘ 3ํ‘ฅ 0 โˆ’3ํ‘ ํ‘’ํ‘›3ํ‘ฅ 1 4 csc 3ํ‘ฅ | = 1 4 csc 3ํ‘ฅํ‘ํ‘œํ‘ 3ํ‘ฅ โˆ’ 0 = 1 4 1 ํ‘ ํ‘’ํ‘›3ํ‘ฅ ํ‘ํ‘œํ‘ 3ํ‘ฅ = 1 4 ํ‘ํ‘กํ‘”3ํ‘ฅ ํ‘ข1 โ€ฒ = 1 4 3 โˆ’ = โˆ’1 12 ํ‘ข1 = โˆซ โˆ’1 12 ํ‘‘ํ‘ฅ = โˆ’1 12 ํ‘ฅ + ํ‘
ํ‘ข2 โ€ฒ = 1 4 ํ‘ํ‘กํ‘”3ํ‘ฅ 3 = 1 12 ํ‘ํ‘œํ‘ 3ํ‘ฅ ํ‘ ํ‘’ํ‘›3ํ‘ฅ ํ‘ข2 = โˆซ 1 12 ํ‘ํ‘กํ‘”3ํ‘ฅํ‘‘ํ‘ฅ 1 36 ํ‘™ํ‘›|ํ‘ ํ‘’ํ‘›3ํ‘ฅ| + ํถ ํ‘ฆํ‘ = ํ‘ฆ1 โˆ— ํ‘ข1 + ํ‘ฆ2 โˆ— ํ‘ข2 = ํ‘ํ‘œํ‘ 3ํ‘ฅ โˆ— (โˆ’ 1 12 ํ‘ฅ) + ํ‘ ํ‘’ํ‘›3ํ‘ฅ โˆ— 1 36 ํ‘™ํ‘›|ํ‘ ํ‘’ํ‘›3ํ‘ฅ| ํ‘ฆ = ํ‘1ํ‘ํ‘œํ‘ 3ํ‘ฅ + ํ‘2ํ‘ ํ‘’ํ‘›3ํ‘ฅ โˆ’ 1 12 ํ‘ฅ โˆ— ํ‘ํ‘œํ‘ 3ํ‘ฅ + 1 36 . ํ‘ ํ‘’ํ‘›3ํ‘ฅ โˆ— ํ‘™ํ‘›|ํ‘ ํ‘’ํ‘›3ํ‘ฅ|

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  • 1. UNIVERSIDAD FERMIN TORO VICERECTORADO ACADEMICO DECANATO DE INGENIERIA MATEMATICA IV ACTIVIDAD EJERCICIOS Carlos Gonzรกlez 24538470
  • 2. 1.) Determine si la funciรณn es soluciรณn de la ecuaciรณn diferencial. a y ๏€ฝ senx ๏€จ x ๏€ซ ctgx๏€ฉ y ๏€ซ y ๏€ฝ ๏€ญctgx ,, ) ln csc ; 1) ํ‘ฆ = ํ‘ ํ‘’ํ‘› ํ‘ฅln (csc ํ‘ฅ + ํ‘ํ‘œํ‘กํ‘” ํ‘ฅ) ํ‘ฆยด = ํ‘ํ‘œํ‘ ํ‘ฅ ln(csc ํ‘ฅ + ํ‘ํ‘œํ‘กํ‘” ํ‘ฅ) + 1 csc ํ‘ฅ + ํ‘ํ‘œํ‘กํ‘” ํ‘ฅ (โˆ’ csc ํ‘ฅํ‘ํ‘œํ‘กํ‘” ํ‘ฅ) โˆ’ csc 2 ํ‘ฅ ํ‘ ํ‘’ํ‘› ํ‘ฅ ํ‘ฆยด = ํ‘ํ‘œํ‘ ํ‘ฅ ln(csc ํ‘ฅ + ํ‘ํ‘œํ‘กํ‘” ํ‘ฅ) + 1 csc ํ‘ฅ + ํ‘ํ‘œํ‘กํ‘” ํ‘ฅ (โˆ’ csc ํ‘ฅ)(ํ‘ํ‘œํ‘กํ‘” ํ‘ฅ +csc ํ‘ฅ)ํ‘ ํ‘’ํ‘› ํ‘ฅ ํ‘ฆยด = ํ‘ํ‘œํ‘ ํ‘ฅ ln(csc ํ‘ฅ + ํ‘ํ‘œํ‘กํ‘” ํ‘ฅ) โˆ’ csc ํ‘ฅ ํ‘ ํ‘’ํ‘› ํ‘ฅ ํ‘ฆยด = ํ‘ํ‘œํ‘ ํ‘ฅ ln(csc ํ‘ฅ + ํ‘ํ‘œํ‘กํ‘” ํ‘ฅ) โˆ’ 1 ํ‘ฆยดยด = โˆ’ํ‘ ํ‘’ํ‘› ํ‘ฅ ln(csc ํ‘ฅ + ํ‘ํ‘œํ‘กํ‘” ํ‘ฅ) + 1 csc ํ‘ฅ + ํ‘ํ‘œํ‘กํ‘” ํ‘ฅ (โˆ’ csc ํ‘ฅํ‘ํ‘œํ‘กํ‘” ํ‘ฅ โˆ’ csc 2 ํ‘ฅ) ํ‘ํ‘œํ‘  ํ‘ฅ ํ‘ฆยดยด = โˆ’ํ‘ ํ‘’ํ‘› ํ‘ฅ ln(csc ํ‘ฅ + ํ‘ํ‘œํ‘กํ‘” ํ‘ฅ) + 1 csc ํ‘ฅ + ํ‘ํ‘œํ‘กํ‘” ํ‘ฅ (โˆ’ csc ํ‘ฅ)(ํ‘ํ‘œํ‘กํ‘” ํ‘ฅ +csc ํ‘ฅ)ํ‘ํ‘œํ‘  ํ‘ฅ ํ‘ฆยดยด = โˆ’ํ‘ ํ‘’ํ‘› ํ‘ฅ ln(csc ํ‘ฅ + ํ‘ํ‘œํ‘กํ‘” ํ‘ฅ) โˆ’ csc ํ‘ฅ cos ํ‘ฅ ํ‘ฆยดยด = โˆ’ํ‘ ํ‘’ํ‘› ํ‘ฅ ln(csc ํ‘ฅ + ํ‘ํ‘œํ‘กํ‘” ํ‘ฅ) โˆ’ 1 ํ‘ ํ‘’ํ‘›ํ‘ฅ ํ‘ํ‘œํ‘  ํ‘ฅ ํ‘ฆยดยด = โˆ’ํ‘ ํ‘’ํ‘› ํ‘ฅ ln(csc ํ‘ฅ + ํ‘ํ‘œํ‘กํ‘” ํ‘ฅ) โˆ’ ํ‘ํ‘œํ‘กํ‘”ํ‘ฅ ํ‘ฆยดยด + ํ‘ฆ = โˆ’ํ‘ํ‘œํ‘กํ‘” ํ‘ฅ โˆ’ํ‘ ํ‘’ํ‘› ํ‘ฅ ln(csc ํ‘ฅ + ํ‘ํ‘œํ‘กํ‘” ํ‘ฅ) โˆ’ ํ‘ํ‘œํ‘กํ‘”ํ‘ฅ + ํ‘ ํ‘’ํ‘› ํ‘ฅ ln(csc ํ‘ฅ + ํ‘ํ‘œํ‘กํ‘” ํ‘ฅ) = โˆ’ํ‘ํ‘œํ‘กํ‘” ํ‘ฅ โˆ’ํ‘ํ‘œํ‘กํ‘” ํ‘ฅ = โˆ’ํ‘ํ‘œํ‘กํ‘”ํ‘ฅ
  • 3. 2) Resolver las siguientes ecuaciones diferenciales de primer orden de acuerdo al mรฉtodo correspondiente. a .) senx cos xy y tg x , 2 ๏€ซ ๏€ฝ ) ๏€จ 2 y cos ๏€ฉ ๏€จ2 2 y cos 2 ๏€ฉ 0 b e y xy dx xe x xy y dy ๏€ญ ๏€ซ ๏€ญ ๏€ซ ๏€ฝ 2- a) ํ‘ ํ‘’ํ‘›ํ‘ฅ cos ํ‘ฅ ํ‘‘ํ‘ฆ ํ‘‘ํ‘ฅ + ํ‘ฆ = ํ‘กํ‘”2ํ‘ฅ ํ‘ ํ‘’ํ‘›ํ‘ฅ cos ํ‘ฅ ํ‘‘ํ‘ฆ ํ‘‘ํ‘ฅ = (ํ‘กํ‘”2ํ‘ฅ โˆ’ ํ‘ฆ) ํ‘ ํ‘’ํ‘›ํ‘ฅ cos ํ‘ฅํ‘‘ํ‘ฆ = (ํ‘กํ‘”2ํ‘ฅ โˆ’ ํ‘ฆ)ํ‘‘ํ‘ฅ ํ‘ ํ‘’ํ‘›ํ‘ฅ cos ํ‘ฅ ํ‘‘ํ‘ฆ + (ํ‘ฆโˆ’ํ‘กํ‘”2ํ‘ฅ)ํ‘‘ํ‘ฅ = 0 ํ‘€ = ํ‘ฆโˆ’ํ‘กํ‘”2ํ‘ฅ ํ‘€ํ‘ฆ = 1 ํ‘ = ํ‘ ํ‘’ํ‘›ํ‘ฅ cos ํ‘ฅ ํ‘ํ‘ฅ = ํ‘ํ‘œํ‘ ํ‘ฅ โˆ— ํ‘ํ‘œํ‘ ํ‘ฅ + ํ‘ ํ‘’ํ‘›ํ‘ฅ โˆ— (โˆ’ํ‘ ํ‘’ํ‘›ํ‘ฅ) = ํ‘ํ‘œํ‘ 2ํ‘ฅ โˆ’ ํ‘ ํ‘’ํ‘›2ํ‘ฅ ํ‘€ํ‘ฆ โˆ’ ํ‘ํ‘ฅ ํ‘ = 1 โˆ’ ํ‘ํ‘œํ‘ 2ํ‘ฅ + ํ‘ ํ‘’ํ‘›2ํ‘ฅ ํ‘ ํ‘’ํ‘› ํ‘ฅ cos ํ‘ฅ = 2ํ‘ ํ‘’ํ‘›2ํ‘ฅ ํ‘ ํ‘’ํ‘› ํ‘ฅ cos ํ‘ฅ = 2ํ‘ ํ‘’ํ‘›ํ‘ฅ cos ํ‘ฅ = 2ํ‘กํ‘”ํ‘ฅ ํ‘’ โˆซ 2ํ‘กํ‘”ํ‘ฅํ‘‘ํ‘ฅ = ํ‘’ln(ํ‘ ํ‘’ํ‘ํ‘ฅ)2 = (ํ‘ ํ‘’ํ‘ํ‘ฅ)2 (ํ‘ ํ‘’ํ‘ํ‘ฅ)2ํ‘ ํ‘’ํ‘›ํ‘ฅcos ํ‘ฅ ํ‘‘ํ‘ฆ + (ํ‘ ํ‘’ํ‘ํ‘ฅ)2(ํ‘ฆโˆ’ํ‘กํ‘”2ํ‘ฅ)ํ‘‘ํ‘ฅ = 0 ํ‘กํ‘”ํ‘ฅ ํ‘‘ํ‘ฆ + (ํ‘ฆํ‘ ํ‘’ํ‘2ํ‘ฅ โˆ’ ํ‘ ํ‘’ํ‘2ํ‘ฅํ‘กํ‘”2ํ‘ฅ)ํ‘‘ํ‘ฅ = 0 ํ‘กํ‘”ํ‘ฅํ‘‘ํ‘ฆ + (ํ‘ฆํ‘ ํ‘’ํ‘2ํ‘ฅ โˆ’ ํ‘ ํ‘’ํ‘2ํ‘ฅํ‘กํ‘”2ํ‘ฅ)ํ‘‘ํ‘ฅ = 0 ํ‘€ = ํ‘ฆํ‘ ํ‘’ํ‘2ํ‘ฅ โˆ’ ํ‘ ํ‘’ํ‘2ํ‘ฅํ‘กํ‘”2ํ‘ฅ ํ‘€ํ‘ฆ = ํ‘ ํ‘’ํ‘2ํ‘ฅ ํ‘ = ํ‘กํ‘”ํ‘ฅ ํ‘ํ‘ฅ = sec 2 ํ‘ฅ
  • 4. โˆซ ํ‘‘ํ‘“ = โˆซ ํ‘€ํ‘‘ํ‘ฅ โˆซ ํ‘‘ํ‘“ = โˆซ(ํ‘ฆํ‘ ํ‘’ํ‘2ํ‘ฅ โˆ’ ํ‘ ํ‘’ํ‘2ํ‘ฅํ‘กํ‘”2ํ‘ฅ)ํ‘‘ํ‘ฅ ํ‘“ = ํ‘ฆํ‘กํ‘”ํ‘ฅ โˆ’ ํ‘กํ‘”3 3 + โ„Ž(ํ‘ฆ) ํ‘“ํ‘ฆ = ํ‘กํ‘”ํ‘ฅ โˆ’ โ„Žโ€ฒ(ํ‘ฆ) ํ‘“ํ‘ฆ = ํ‘ ํ‘กํ‘”ํ‘ฅ + โ„Žโ€ฒ(ํ‘ฆ) = ํ‘กํ‘”ํ‘ฅ โˆซ ํ‘‘โ„Ž(ํ‘ฆ) = โˆซ 0ํ‘‘ํ‘ฆ โ„Ž(ํ‘ฆ) = ํถ ํ‘“ = ํ‘ฆํ‘กํ‘”ํ‘ฅ + ํ‘กํ‘”3ํ‘ฅ 3 + ํ‘ 2.b-) (ํ‘’2ํ‘ฆ โˆ’ ํ‘ฆํ‘ํ‘œํ‘  ํ‘ฅํ‘ฆ)ํ‘‘ํ‘ฅ + (2ํ‘ฅํ‘’2ํ‘ฆ โˆ’ ํ‘ฅํ‘ํ‘œํ‘  ํ‘ฅํ‘ฆ + 2ํ‘ฆ)ํ‘‘ํ‘ฆ = 0 ํ‘€(ํ‘ฅ, ํ‘ฆ) = ํ‘’2ํ‘ฆ โˆ’ ํ‘ฆํ‘ํ‘œํ‘  ํ‘ฅํ‘ฆ ํ‘(ํ‘ฅ, ํ‘ฆ) = 2ํ‘ฅํ‘’2ํ‘ฆ โˆ’ ํ‘ฅํ‘ํ‘œํ‘  ํ‘ฅํ‘ฆ + 2ํ‘ฆ ํ‘€ํ‘ฆ = 2ํ‘’2ํ‘ฆ โˆ’ 1cos ํ‘ฅํ‘ฆ + (โˆ’ํ‘ฆ)(โˆ’ํ‘ ํ‘’ํ‘› ํ‘ฅํ‘ฆ)ํ‘ฅ = 2ํ‘’2ํ‘ฆ โˆ’ cos ํ‘ฅํ‘ฆ + (ํ‘ฅํ‘ฆํ‘ ํ‘’ํ‘› ํ‘ฅํ‘ฆ) ํ‘ํ‘ฅ = 2ํ‘’2ํ‘ฆ โˆ’ 1 cos ํ‘ฅํ‘ฆ + (โˆ’ํ‘ฅ)(โˆ’ํ‘ ํ‘’ํ‘› ํ‘ฅํ‘ฆ)ํ‘ฆ = 2ํ‘’2ํ‘ฆ โˆ’ cos ํ‘ฅํ‘ฆ + (ํ‘ฅํ‘ฆํ‘ ํ‘’ํ‘› ํ‘ฅํ‘ฆ) ํ‘€ํ‘ฆ = ํ‘ํ‘ฅ ํธํ‘ฅํ‘Žํ‘ํ‘กํ‘Ž ํœ•ํ‘“(ํ‘ฅ, ํ‘ฆ) ํœ•ํ‘ฅ = ํ‘€(ํ‘ฅ, ํ‘ฆ) โˆซ ํœ•ํ‘“(ํ‘ฅ, ํ‘ฆ) = โˆซ(ํ‘’2ํ‘ฆ โˆ’ ํ‘ฆํ‘ํ‘œํ‘  ํ‘ฅํ‘ฆ)ํ‘‘ํ‘ฅ = ํ‘“(ํ‘ฅ, ํ‘ฆ) = ํ‘ฅํ‘’2ํ‘ฆ โˆ’ ํ‘ฆํ‘ ํ‘’ํ‘› ํ‘ฅํ‘ฆ ํ‘ฆ + โ„Ž(ํ‘ฆ) ํ‘“(ํ‘ฅ, ํ‘ฆ) = ํ‘ฅํ‘’2ํ‘ฆ โˆ’ ํ‘ ํ‘’ํ‘› ํ‘ฅํ‘ฆ + โ„Ž(ํ‘ฆ)
  • 5. ํœ•ํ‘“(ํ‘ฅ, ํ‘ฆ) ํœ•ํ‘ฆ = 2ํ‘ฅํ‘’2ํ‘ฆ โˆ’ ํ‘ฅํ‘ํ‘œํ‘  ํ‘ฅํ‘ฆ + โ„Žยด(ํ‘ฆ) ํœ•ํ‘“(ํ‘ฅ, ํ‘ฆ) ํœ•ํ‘ฆ = ํ‘(ํ‘ฅ, ํ‘ฆ) 2ํ‘ฅํ‘’2ํ‘ฆ โˆ’ ํ‘ฅํ‘ํ‘œํ‘  ํ‘ฅํ‘ฆ + โ„Žยด(ํ‘ฆ) = 2ํ‘ฅํ‘’2ํ‘ฆ โˆ’ ํ‘ฅํ‘ํ‘œํ‘  ํ‘ฅํ‘ฆ + 2ํ‘ฆ โ„Žยด(ํ‘ฆ) = 2ํ‘ฆ 2 โˆซ โ„Ž(ํ‘ฆ)ํ‘‘ํ‘ฆ โ„Ž(ํ‘ฆ) = ํ‘ฆ2 + ํถ ํ‘“(ํ‘ฅ, ํ‘ฆ) = ํ‘ฅํ‘’2ํ‘ฆ โˆ’ ํ‘ ํ‘’ํ‘› ํ‘ฅํ‘ฆ + ํ‘ฆ2 + ํถ 3.) Resolver las ecuaciones diferenciales de orden N por coeficientes indeterminados. a .) y ,, ๏€ซ y , ๏€ฝ 2 e 2 x senx .) ,, 9 93 3cos b y ๏€ซ y ๏€ฝ x ๏€ซ x 3 a) ํ‘ฆยดยด + ํ‘ฆยด = 2ํ‘’2ํ‘ฅ ํ‘ ํ‘’ํ‘› ํ‘ฅ Para la soluciรณn homogenea ํ‘š2 + ํ‘š = 0 ํ‘š = 0 ํ‘œ ํ‘š = โˆ’1 ํ‘ฆโ„Ž = ํ‘1 + ํ‘2ํ‘’โˆ’ํ‘ฅ ํท(ํท + 1)(ํท2 โˆ’ 4ํท + 5)ํ‘ฆ = 0 ํท(ํท + 1)(ํท โˆ’ 2 โˆ’ ํ‘–)(ํท โˆ’ 2 + ํ‘–)ํ‘ฆ = 0 ํ‘ฆ = ํ‘1 + ํ‘2ํ‘’โˆ’ํ‘ฅ + ํดํ‘’2ํ‘ฅ ํ‘ํ‘œํ‘ ํ‘ฅ + ํตํ‘’2ํ‘ฅ ํ‘ ํ‘’ํ‘›ํ‘ฅ
  • 6. Yh yp ํ‘ฆยดํ‘ = ํด2ํ‘’2ํ‘ฅ ํ‘ํ‘œํ‘ ํ‘ฅ + (โˆ’ํ‘ ํ‘’ํ‘›ํ‘ฅ)ํดํ‘’2ํ‘ฅ + ํต2ํ‘’2ํ‘ฅ ํ‘ ํ‘’ํ‘›ํ‘ฅ + ํ‘ํ‘œํ‘ ํ‘ฅํตํ‘’2ํ‘ฅ = (2ํด + ํต)ํ‘’2ํ‘ฅ ํ‘ํ‘œํ‘ ํ‘ฅ + (โˆ’ํด + 2ํต)ํ‘’2ํ‘ฅ ํ‘ ํ‘’ํ‘›ํ‘ฅ ํ‘ฆยดยดํ‘ = 2(2ํด + ํต)ํ‘’2ํ‘ฅ ํ‘ํ‘œํ‘ ํ‘ฅ + (2ํด + ํต)ํ‘’2ํ‘ฅ (โˆ’ํ‘ ํ‘’ํ‘›ํ‘ฅ) + 2(โˆ’ํด + 2ํต)ํ‘’2ํ‘ฅ ํ‘ ํ‘’ํ‘›ํ‘ฅ + (โˆ’ํด + 2ํต)ํ‘’2ํ‘ฅ ํ‘ํ‘œํ‘ ํ‘ฅ = (3ํด + ํต)ํ‘’2ํ‘ฅ ํ‘ํ‘œํ‘ ํ‘ฅ + (โˆ’4ํด + 3ํต)ํ‘’2ํ‘ฅ ํ‘ ํ‘’ํ‘›ํ‘ฅ ํ‘ฆยดยดํ‘ + ํ‘ฆยดํ‘ = (5ํด + 5ํต)ํ‘’2ํ‘ฅ ํ‘ํ‘œํ‘ ํ‘ฅ + (โˆ’5ํด + 5ํต)ํ‘’2ํ‘ฅ ํ‘ ํ‘’ํ‘›ํ‘ฅ = 2ํ‘’2ํ‘ฅ ํ‘ ํ‘’ํ‘›ํ‘ฅ โˆ’5ํด + 5ํต = 2 5ํด + 5ํต = 0 ํด = โˆ’ 1 5 ํ‘ฆ ํต = 1 5 ํ‘ฆ = ํ‘1 + ํ‘2ํ‘’โˆ’ํ‘ฅ โˆ’ 1 5 ํ‘’2ํ‘ฅ ํ‘ํ‘œํ‘ ํ‘ฅ + 1 5 ํ‘’2ํ‘ฅ ํ‘ ํ‘’ํ‘›ํ‘ฅ 3 b ) ํ‘ฆยดยด + 9ํ‘ฆ = 93ํ‘ฅ + 3ํ‘ํ‘œํ‘ ํ‘ฅ (ํท2 + 9)ํท2(ํท2 + 1)ํ‘ฆ = 0
  • 7. ํ‘ฆ = ํถ1ํ‘ํ‘œํ‘ 3ํ‘ฅ + 2ํถํ‘ ํ‘’ํ‘›3ํ‘ฅ + ํถ3 + ํถ4ํ‘ฅ + ํถ5ํ‘ํ‘œํ‘ ํ‘ฅ + ํถ6ํ‘ ํ‘’ํ‘›ํ‘ฅ Yh yp ํ‘ฆยดํ‘ = ํถ4 โˆ’ ํถ5ํ‘ ํ‘’ํ‘›ํ‘ฅ + ํถ6ํ‘ํ‘œํ‘ ํ‘ฅ ํ‘ฆยดยดํ‘ = โˆ’ํถ5ํ‘ํ‘œํ‘ ํ‘ฅ โˆ’ ํถ6ํ‘ ํ‘’ํ‘›ํ‘ฅ ํ‘ฆยดยดํ‘ + 9ํ‘ฆํ‘ = 93ํ‘ฅ + 3ํ‘ํ‘œํ‘ ํ‘ฅ โˆ’ํถ5ํ‘ํ‘œํ‘ ํ‘ฅ โˆ’ ํถ6ํ‘ ํ‘’ํ‘›ํ‘ฅ + 9(ํถ3 + ํถ4ํ‘ฅ + ํถ5ํ‘ํ‘œํ‘ ํ‘ฅ + ํถ6ํ‘ ํ‘’ํ‘›ํ‘ฅ) = 93ํ‘ฅ + 3ํ‘ํ‘œํ‘ ํ‘ฅ 9ํถ3 + 9ํถ4ํ‘ฅ + 8ํถ5ํ‘ํ‘œํ‘ ํ‘ฅ + 8ํถ6ํ‘ ํ‘’ํ‘›ํ‘ฅ = 93ํ‘ฅ + 3ํ‘ํ‘œํ‘ ํ‘ฅ 9ํถ3 = 0 ํถ3 = 0 9ํถ4 = 93 ํถ4 = 93 9 8ํถ5 = 3 ํถ5 = 3 8 8ํถ6 = 0 ํถ6 = 0 ํ‘ฆ = ํถ1ํ‘ํ‘œํ‘ 3ํ‘ฅ + 2ํถํ‘ ํ‘’ํ‘›3ํ‘ฅ + 0 + ํถ4ํ‘ฅ + ํถ5ํ‘ํ‘œํ‘ ํ‘ฅ + 0ํ‘ ํ‘’ํ‘›ํ‘ฅ ํ‘ฆ = ํถ1ํ‘ํ‘œํ‘ 3ํ‘ฅ + ํถ2ํ‘ ํ‘’ํ‘›3ํ‘ฅ + 93 9 ํ‘ฅ + 3 8 ํ‘ํ‘œํ‘ ํ‘ฅ
  • 8. 4.) Resuelva por variaciรณn de parรกmetros: 1 9 ,, ๏€ซ ๏€ฝ 4 y y Csc3x 4 a) ํ‘ฆยดยด + 9ํ‘ฆ = 1 4 csc 3ํ‘ฅ ํ‘š2 + 9 = 0 ํ‘ฆโ„Ž = ํ‘1ํ‘ํ‘œํ‘ 3ํ‘ฅ + ํ‘2ํ‘ ํ‘’ํ‘›3ํ‘ฅ y1 y2 ํ‘ฆ1โ€ฒ = โˆ’3ํ‘ ํ‘’ํ‘›3ํ‘ฅ ํ‘ฆ2โ€ฒ = 3ํ‘ํ‘œํ‘ 3ํ‘ฅ ํ‘“(ํ‘ฅ) = 1 4 csc 3ํ‘ฅ ํ‘ค = | ํ‘ํ‘œํ‘ 3ํ‘ฅ ํ‘ ํ‘’ํ‘›3ํ‘ฅ โˆ’3ํ‘ ํ‘’ํ‘›3ํ‘ฅ 3ํ‘ํ‘œํ‘ 3ํ‘ฅ | = 3ํ‘ํ‘œํ‘ 23ํ‘ฅ + 3ํ‘ ํ‘’ํ‘›23ํ‘ฅ = 3 ํ‘ค1 = | 0 ํ‘ ํ‘’ํ‘›3ํ‘ฅ 1 4 csc 3ํ‘ฅ 3ํ‘ํ‘œํ‘ 3ํ‘ฅ | = 0 โˆ’ 1 4 csc 3ํ‘ฅํ‘ ํ‘’ํ‘›3ํ‘ฅ = โˆ’ 1 4 1 ํ‘ ํ‘’ํ‘›3ํ‘ฅ ํ‘ ํ‘’ํ‘›3ํ‘ฅ = โˆ’ 1 4 ํ‘ค2 = | ํ‘ํ‘œํ‘ 3ํ‘ฅ 0 โˆ’3ํ‘ ํ‘’ํ‘›3ํ‘ฅ 1 4 csc 3ํ‘ฅ | = 1 4 csc 3ํ‘ฅํ‘ํ‘œํ‘ 3ํ‘ฅ โˆ’ 0 = 1 4 1 ํ‘ ํ‘’ํ‘›3ํ‘ฅ ํ‘ํ‘œํ‘ 3ํ‘ฅ = 1 4 ํ‘ํ‘กํ‘”3ํ‘ฅ ํ‘ข1 โ€ฒ = 1 4 3 โˆ’ = โˆ’1 12 ํ‘ข1 = โˆซ โˆ’1 12 ํ‘‘ํ‘ฅ = โˆ’1 12 ํ‘ฅ + ํ‘
  • 9. ํ‘ข2 โ€ฒ = 1 4 ํ‘ํ‘กํ‘”3ํ‘ฅ 3 = 1 12 ํ‘ํ‘œํ‘ 3ํ‘ฅ ํ‘ ํ‘’ํ‘›3ํ‘ฅ ํ‘ข2 = โˆซ 1 12 ํ‘ํ‘กํ‘”3ํ‘ฅํ‘‘ํ‘ฅ 1 36 ํ‘™ํ‘›|ํ‘ ํ‘’ํ‘›3ํ‘ฅ| + ํถ ํ‘ฆํ‘ = ํ‘ฆ1 โˆ— ํ‘ข1 + ํ‘ฆ2 โˆ— ํ‘ข2 = ํ‘ํ‘œํ‘ 3ํ‘ฅ โˆ— (โˆ’ 1 12 ํ‘ฅ) + ํ‘ ํ‘’ํ‘›3ํ‘ฅ โˆ— 1 36 ํ‘™ํ‘›|ํ‘ ํ‘’ํ‘›3ํ‘ฅ| ํ‘ฆ = ํ‘1ํ‘ํ‘œํ‘ 3ํ‘ฅ + ํ‘2ํ‘ ํ‘’ํ‘›3ํ‘ฅ โˆ’ 1 12 ํ‘ฅ โˆ— ํ‘ํ‘œํ‘ 3ํ‘ฅ + 1 36 . ํ‘ ํ‘’ํ‘›3ํ‘ฅ โˆ— ํ‘™ํ‘›|ํ‘ ํ‘’ํ‘›3ํ‘ฅ|