The homogenization method: an introduction (1993) by Lars Erik Persson, etc.
By Lars Erik Persson, etc.
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Additional info for The homogenization method: an introduction (1993)
Example text
Log | cosec x − cotg x| + C = log tg sen x 2 (k) sh x dx = ch x + C. (l) ch x dx = sh x + C. (m) dx = th x + C. ch2 x (n) dx = − coth x + C. sh2 x (˜ n) 1 x 1 x dx = arc tg + C = − arc cotg + C1 , x2 + a 2 a a a a (o) (p) a > 0. x2 dx 1 x−a + C, = log 2 −a 2a x+a a2 1 1 x a+x dx + C = arg th + C, = log 2 −x 2a a−x a a + C. a = 0. a = 0. a = 0. (q) √ dx = log x + x2 + a x2 + a + C = 1 x arg sh √ + C, a a a > 0. (r) √ dx = log x + x2 − a x2 − a + C = 1 x arg ch √ + C, a a a > 0. (s) √ a2 x x dx = arc sen + C = − arc cos + C1 , 2 a a −x a > 0.
Cn−1 ) dx φ(x, C1 , . . , Cn−1 ) dx + Cn yn (x), donde Cn surge como constante de integraci´on al calcular una primitiva de φ (que aparece representada con la misma notaci´on de integral, esperando que este peque˜ no y habitual abuso de notaci´on no introduzca confusi´on en el lector). Si no conseguimos resolverla en general, se puede intentar reducir el orden de nuevo encontrando ahora un−1 (x) soluci´ on particular de la homog´enea asociada. Si este proceso lo logramos hacer el suficiente n´ umero de veces, llegaremos siempre a una ecuaci´on lineal de orden 1, que s´ı que sabemos resolver.
Lineal general de orden n > 1 no es soluble por cuadraturas. S´ı que pueden resolverse las ecuaciones lineales de orden n con coeficientes constantes; aunque ´este es un tema central en el estudio de la teor´ıa de E. D. , no nos ocuparemos de ello en estas notas. Realmente, lo u ´nico que vamos a ver aqu´ı es una serie de m´etodos que permiten reducir el orden de una E. D. Aplic´andolos (sucesivamente si es necesario), podremos llegar a una ecuaci´on de primer orden que, con un poco de suerte, se encontrar´a entre las que ya sabemos resolver.