Bsp 51

[dose]

...war mir dann nachm 5. Mal falsch rechnen scho zdeppat, das Gleichungssystem mit der Hand zu lösen ;)

> ode := diff(y(x),x,x)+2*diff(y(x),x)+y(x) = 0;

/ 2 \
|d | /d \
ode := |--- y(x)| + 2 |-- y(x)| + y(x) = 0
| 2 | \dx /
\dx /

> dsolve(ode);

y(x) = _C1 exp(-x) + _C2 exp(-x) x

> g1 := _C1*exp(-x) + _C2*exp(-x)*x;
> g2 := _D1*exp(-x) + _D2*exp(-x)*x;


g1 := _C1 exp(-x) + _C2 exp(-x) x


g2 := _D1 exp(-x) + _D2 exp(-x) x

> g1_1 := diff(g1,x);
> g2_1 := diff(g2,x);

g1_1 := -_C1 exp(-x) - _C2 exp(-x) x + _C2 exp(-x)


g2_1 := -_D1 exp(-x) - _D2 exp(-x) x + _D2 exp(-x)

> GL1 := simplify(subs(x=0,g1_1)+subs(x=1,g2_1)=0);
> GL2 := simplify(subs(x=0,g1)+subs(x=1,g2)+subs(x=0,g1_1)= 0);
> GL3 := simplify(subs(x=w,g1) = subs(x=w,g2));
> GL4 := simplify(subs(x=w,g2_1) - subs(x=w,g1_1) = 1);

GL1 := -_C1 + _C2 - _D1 exp(-1) = 0


GL2 := _D1 exp(-1) + _D2 exp(-1) + _C2 = 0


GL3 := _C1 exp(-w) + _C2 exp(-w) w = _D1 exp(-w) + _D2 exp(-w) w


GL4 := -_D1 exp(-w) - _D2 exp(-w) w + _D2 exp(-w) + _C1 exp(-w)

+ _C2 exp(-w) w - _C2 exp(-w) = 1

> L := simplify(solve({GL1,GL2,GL3,GL4}));

exp(w - 1) (2 w - 1 + w exp(-1))
L := {_C1 = --------------------------------,
%1

exp(w - 1) (exp(-1) - w + 1)
_C2 = - ----------------------------, w = w,
%1

exp(w) (2 exp(-1) + w exp(-1) + 1)
_D2 = ----------------------------------,
%1

(exp(-1) + w exp(-1) + w) exp(w)
_D1 = - --------------------------------}
%1

%1 := 3 exp(-1) + 1 + exp(-2)

> simplify(subs(L,g1));

exp(w - 1 - x) (-2 w + 1 - w exp(-1) + x exp(-1) - x w + x)
- -----------------------------------------------------------
3 exp(-1) + 1 + exp(-2)

> simplify(subs(L,g2));

exp(w - x)

(-exp(-1) - w exp(-1) - w + 2 x exp(-1) + x w exp(-1) + x)/(

3 exp(-1) + 1 + exp(-2))