Jan Math 2008-12-05

allgemein

homogene DGL
$ay''+by'+cy=0$ (oder auch höhere Ableitungen von y)
inhomogene DGL
$ay''+by'+cy=f(x)$
Gelöst witd zuerst die homogene DGL - die Lösung der inhomogenen ist eine (irgend eine) Löung der inhomogenen plus die allgemeine Löung der homogenen
$y=y_{sp}+y_{h}$

- 4.1 d)

$y''+6x=0$
- homogene -
$y''=0$
D.h. y zweimal differenziert ist 0, da kann y maximal x sein (Polynom ersten Grades). Homogene Lösung (allgemein)
$y_{h}=ax+b$
- spezielle Lösung -
$y_{sp}''=-6x$
Einfach zweimal integrieren:
$y_{sp}'=-3x^{2}$
(kein +C, da man ja nur eine spezielle Lösung sucht!)
$y_{sp}=-x^{3}$
- Gesamtlösung -
$y=y_{sp}+y_{h}$
$y=-x^{3}+ax+b$
(a,b beliebig)

- 4.1 e)

$y''+6x-3=0$
- homogene -
$y_{h}''=0$
$y_{h}=ax+b$
- spezielle -
$y_{sp}''=-6x+3$
$y_{sp}'=-3x^{2}+3x$
$y_{sp}=-x^{3}+{3 \over 2}x^{2}$
- Gesamtlösung -
$y=y_{sp}+y_{h}$
$y=-x^{3}+{3 \over 2}x^{2}+ax+b$

- 4.1 f)

$y''+12x^{2}-4x=1$
- homogene (wie schon zwei Mal) -
$y_{h}=ax+b$
- spezeille -
$y_{sp}''=-12x^{2}+4x+1$
$y_{sp}'=-4x^{3}+2x^{2}+x$
$y_{sp}=-x^{4}+{2 \over 3}x^{3}+{1 \over 2}x^{2}$
$y_{sp2}=-x^{4}+{2 \over 3}x^{3}$
- Gesmatlösung -
$y=-x^{4}+{2 \over 3}x^{3}+{1 \over 2}x^{2}+ax+b$

- 4.2 c)

$y''-x+1=0$
$y(1)=0$
$y'(1)=0$
- allgemein -
$y''=x-1$
$y_{h}''=0$
$y_{h}=ax+b$
$y_{sp}''=x-1$
$y_{sp}'={x^{2} \over 2}-x$
$y_{sp}={x^{3} \over 6}-{x^{2} \over 2}$
$y={x^{3} \over 6}-{x^{2} \over 2}+ax+b$ [1]
$y'={x^{2} \over 2}-x+a$ [2]
Jetzt in [2] laut Anfangsbedingung einsetzen:
$0={1 \over 2}-1+a$
$a={1 \over 2}$
Jetzt in [1] laut Anfangsbedingung einsetzen:
$0={1 \over 6}-{1 \over 2}+a+b$
$0={1 \over 6}-{1 \over 2}+{1 \over 2}+b$
$b=-{1 \over 6}$
$y={x^{3} \over 6}-{x^{2} \over 2}+{1 \over 2}x-{1 \over 6}$

- 4.2 d)

$y''+6x^{2}=1-x$
$y(0)=2$
$y'(0)=5$
- allgemein -
$y''=-6x^{2}-x+1$
$y_{h}=ax+b$
$y_{sp}''=-6x^{2}-x+1$
$y_{sp}'=-2x^{3}-{1 \over 2}x^{2}+x$
$y_{sp}=-{1 \over 2}x^{4}-{1 \over 6}x^{3}+{1 \over 2}x^{2}$
$y=-{1 \over 2}x^{4}-{1 \over 6}x^{3}+{1 \over 2}x^{2}+ax+b$ [1]
$y'=-2x^{3}-{1 \over 2}x^{2}+x+a$ [2]
Einsetzen in [2]
$5=0+0+0+a$
$a=5$
Einsetzen in [1]
$2=0+0+0+0+b$
$b=2$
$y=-{1 \over 2}x^{4}-{1 \over 6}x^{3}+{1 \over 2}x^{2}+5x+2$

- 4.3 a)

$y''=2x-1$
$y(0)=1$
$y(6)=1$
$y_{h}=ax+b$
$y_{sp}''=2x-1$
$y_{sp}'=x^{2}-x$
$y_{sp}={1 \over 3}x^{3}-{1 \over 2}x^{2}$
$y={1 \over 3}x^{3}-{1 \over 2}x^{2}+ax+b$
Einsetzen y(0)=1
$1=0-0+0+b$
$b=1$
Einsetzen y6)=1
$1={1 \over 3}6^{3}-{1 \over 2}6^{2}+a6+b$
$1=72-18+6a+1$
$-54=6a$
$a=9$
$y={1 \over 3}x^{3}-{1 \over 2}x^{2}+9x+1$

- 4.3 b)

$y''-x+3=0$
$y(3)=1$
$y(9)=10$
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_h=ax+b}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_{sp}''=x-3}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_{sp}'={1 \over 2} x^2 - 3x}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_{sp}={1 \over 6} x^3 - {3 \over 2} x^2}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y={1 \over 6} x^3 - {3 \over 2} x^2+ax+b}
Einsetzen => lineares Gleichungssystem:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1={1 \over 6} 3^3 - {3 \over 2} 3^2+a3+b}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1={27 \over 6} - {27 \over 2}+ 3a+b}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3a+b={6 \over 6} - {27 \over 6} + {81 \over 6}={60 \over 6}=10} [1]
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1={1 \over 6} 9^3 - {3 \over 2} 9^2+a9+b}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1={729 \over 6} - {243 \over 2} + 9a+b}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 9a+b={6 \over 6} - {729 \over 6} + {729 \over 6}=1} [2]
Subtrahiere [1] von [2]
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 6a=1-10=-9}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a=-{9 \over 6}=-{3 \over 2}}
Einsetzen in [1]
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3(-{3 \over 2})+b=10}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b=10+{9 \over 2}={29 \over 2}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y={1 \over 6} x^3 - {3 \over 2} x^2-{3 \over 2}x+{29 \over 2}}

- 4.4

Vorläufig aufgeschoben... Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M_b}  ? Biegung ?

-4.7 g)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y'-y^2cosx=0}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y'=y^2cosx}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {dy \over dx} =y^2cosx} (* dx / Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y^2} )
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {1 \over {y^2}} dy = cosx dx} (integrieren)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {-1 \over y} = sinx +c}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y={-1 \over {sinx + c}}}
Probe:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y'={ cosx \over {(sinx +c)^2}}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y^2={1 \over {(sinx +c)^2}}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y'=y^2 cosx} passt

- 4.7 h)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y' cosx + y sinx = 0}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y' cosx = -y sinx}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {dy \over dx} cosx = -y sinx} (* dx / cosx / y)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {1 \over y} dy = { -sinx \over cosx } dx} (integrieren)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle lny=-ln({2 \over {1 + cosx}}) + ln(-{{2 cosx} \over {1 + cosx}}) +c_1=ln(cosx)+c_1} (e^ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_2=e^{c_1}} )
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = c_2 cosx}
Probe:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y'=c_2 -sinx}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y' cosx = c_2 -sinx cosx}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y sinx = c_2 sinx cosx}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y' cosx + y sinx = c_3 (-sinx cosx + sinx cosx) = 0}

-4.7 i)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y'sinx + y=0}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y'sinx=-y}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {dy \over dx} sinx = -y} (* dx / y / sinx)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {1 \over y} dy = {-1 \over sinx } dx} (integrieren)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle lny = -ln({sinx \over {1 + cosx}}) + c_1} (e^ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_2=e^{c_1}} )
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=c_2 {{1 + cosx} \over sinx}}
Probe:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y'=c_2 {{-sinx sinx - (1+cosx) cosx} \over {sin^2 x}}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y'=c_2{{-sin^2 x -cosx -cos^2 x} \over {sin^2 x}}} (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle sin^2x+cos^2x = 1} )
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y'=c_2{{-1 -cosx} \over {sin^2 x}}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y' sinx = -y} passt

-4.8 g)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x y' + x y = y}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x y' = y (1-x)}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x {dy \over dx} = y (1-x)} (* dx / x / y)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {1 \over y} dy = {{1-x} \over x} dx} (integrieren)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle lny = lnx -x +c_1}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=c_2 {x \over {e^x}}}
Probe:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y'=c_2{{1 e^x - x e^x \over (e^x)^2}}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y'=c_2{{e^x (1-x)} \over {e^x e^x}}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y'=c_2{(1-x) \over e^x}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x y'= c_2{{x(1-x)} \over e^x}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x y = c_2 {x^2 \over e^x}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x y' + x y = c_2 {{x(1-x) + x^2} \over e^x}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x y' + x y = c_2 {{x -x^2 + x^2} \over e^x}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x y' + x y = c_2 {x \over e^x}} passt
Ensetzen y(1)=1
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1=c_2 {1 \over {e^1}}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_2=e}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=e{x \over {e^x}}}