Serie2

> restart;
SERIE 1 (CAPÍTULO II)
SEMESTRE 2012-2

1) OBTENER LA SOLUCIÓN GENERAL DE LA ECUACIÓN DIFERENCIAL SIGUIENTE (SIN UTILIZAR dsolve)
3
/ 3 \ / d \ / 3 \ x - 2
x \x + 1/ |--- y(x)| + \2 x - 1/ y(x) = ------
\ dx / x> Ecu := diff(y(x), x)+(2*x^3-1)*y(x)/x(x^3+1) = (x^3-2)/(x*x(x^3+1));
/ 3 \ 3
/ d \ \2 x - 1/ y(x) x - 2
|--- y(x)| + --------------- = -----------
\ dx / / 3 \ / 3 \
x\x + 1/ x x\x + 1/
> EcuNorm :=expand(simplify(Ecu/x(x^3+1)));
d
--- y(x) 3 2
dx 2 y(x) x y(x) x 2
--------- + ---------- - ---------- = ---------- - ------------
/ 3 \ 2 2 2 2
x\x +1/ / 3 \ / 3 \ / 3 \ / 3 \
x\x + 1/ x\x + 1/ x\x + 1/ x\x + 1/ x
> EcuHom := lhs(EcuNorm) = 0;
d
--- y(x) 3
dx 2 y(x) x y(x)
--------- + ---------- - ---------- = 0/ 3 \ 2 2
x\x + 1/ / 3 \ / 3 \
x\x + 1/ x\x + 1/
> Q(x):=rhs(EcuNorm);
2
x 2
---------- - ------------
22
/ 3 \ / 3 \
x\x + 1/ x\x + 1/ x
> P(x):=(2 y(x)x**3-y(x))/(x(x**3+1));
3
2 y(x) x - y(x)
----------------
/ 3 \
x\x + 1/> SolGenral := C[1]*exp(-(int(P(x), x)))+(exp(-(int(P(x), x))))(int(exp(int(P(x), x))*Q(x), x));
/ / 3 \\ / / / 3 \\\/ /
| |2 y(x) x - y(x) || | | |2 y(x) x - y(x) |||| |
C[1] exp|-int|----------------, x|| + |exp|-int|----------------, x||||int|
| | / 3 \ || | | | / 3 \|||| |
\ \ x\x + 1/ // \ \ \ x\x + 1/ ///| |
\ \

/ / 3 \\ / 2 \ \\
| |2 y(x) x - y(x) || | x 2 | ||
exp|int|----------------, x|| |---------- - ------------|, x||
| | / 3 \|| | 2 2 | ||
\ \ x\x + 1/ // | / 3 \ / 3 \ | ||
\x\x + 1/ x\x + 1/ x/ //
>
> comp := dsolve(Ecu);
// / / 3 \ \ \
|| / | | 2 x - 1 | / 3 \ | |
|| | exp| | --------- dx| \x- 2/ | |
|| | | | / 3 \ | | | / / 3 \
|| | \/ x\x + 1/ / | | | | 2 x - 1 |
y(x) = || | ------------------------------ dx| + _C1| exp| | - --------- dx|
|| | / 3 \ | | | | / 3 \ |
|| | x x\x + 1/...