Certamen 2 - Matemáticas III (2009-2).pdf - Pauta Certamen 2 MAT-023 Segundo Semestre 2009 Departamento de Matem´ aticas UTFSM 2 dy 1 Escribir la

Certamen 2 - Matemáticas III (2009-2).pdf - Pauta...

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Pauta Certamen 2, MAT-023, Segundo Semestre 2009 Departamento de Matem´ aticas, UTFSM 1. Escribir la ecuaci´ on en forma expl´ ıcita dy dx = 1 + y x + y 2 x 2 . Homog´ enea: u = y/x, dy dx = x du dx + u . Sustituir en la ecuaci´ on: x du dx + u = 1 + u + u 2 . 10 puntos Por separables, 1 1+ u 2 du = 1 x dx . Entonces, arctan u = ln | x | + C para una constante C . 5 puntos Despejar: u = tan(ln | x | + C ). Entonces, y = x tan(ln | x | + C ). 5 puntos La condici´ on inicial y ( e ) = 0 implica que C = - 1. Las soluci´ on del problema de valor inicial es y = x tan(ln | x | - 1). 5 puntos (Dominio ] e - π/ 2+1 , e π/ 2+1 [.) 2. Ecuaci´ on caracter´ ıstica s 2 + 6 s + 9 = 0. Ra´ ız doble s 1 , 2 = - 3. Soluciones de la ecuacio´ on lineal homog´ enea asociada y 1 = e - 3 x , y 2 = xe - 3 x . 5 puntos Coeficientes indeterminados. Para tratar el t´ ermino 1 en el lado derecho usamos la conjetura y p = A . Sustituir en la ecuaci´ on y + 6 y + 9 y = 1. Entonces, A = 1 / 9 , y p = 1 / 9. 5 puntos Para tratar el t´ ermino cos x en el lado derecho usamos complejificaci´ on: cos x = e ix .
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