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)

$y''-x+3=0$
$y(3)=1$
$y(9)=10$
$y_{h}=ax+b$
$y_{sp}''=x-3$
$y_{sp}'={1 \over 2}x^{2}-3x$
$y_{sp}={1 \over 6}x^{3}-{3 \over 2}x^{2}$
$y={1 \over 6}x^{3}-{3 \over 2}x^{2}+ax+b$
Einsetzen => lineares Gleichungssystem:
$1={1 \over 6}3^{3}-{3 \over 2}3^{2}+a3+b$
$1={27 \over 6}-{27 \over 2}+3a+b$
$3a+b={6 \over 6}-{27 \over 6}+{81 \over 6}={60 \over 6}=10$ [1]
$1={1 \over 6}9^{3}-{3 \over 2}9^{2}+a9+b$
$1={729 \over 6}-{243 \over 2}+9a+b$
$9a+b={6 \over 6}-{729 \over 6}+{729 \over 6}=1$ [2]
Subtrahiere [1] von [2]
$6a=1-10=-9$
$a=-{9 \over 6}=-{3 \over 2}$
Einsetzen in [1]
$3(-{3 \over 2})+b=10$
$b=10+{9 \over 2}={29 \over 2}$
$y={1 \over 6}x^{3}-{3 \over 2}x^{2}-{3 \over 2}x+{29 \over 2}$

- 4.4

Vorläufig aufgeschoben... $M_{b}$ ? Biegung ?

-4.7 g)

$y'-y^{2}cosx=0$
$y'=y^{2}cosx$
${dy \over dx}=y^{2}cosx$ (* dx / $y^{2}$)
${1 \over {y^{2}}}dy=cosxdx$ (integrieren)
${-1 \over y}=-sinx-c$
$y={1 \over {sinx+c}}$
Probe:
$y'={cosx \over {(sinx+c)^{2}}}$
$y^{2}={1 \over {(sinx+c)^{2}}}$
$y'=y^{2}cosx$ passt

- 4.7 h)

$y'cosx+ysinx=0$
$y'cosx=-ysinx$
${dy \over dx}cosx=-ysinx$ (* dx / cosx / y)
${1 \over y}dy={-sinx \over cosx}dx$ (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}
$y'cosx+ysinx=c_{3}(-sinxcosx+sinxcosx)=0$

-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}
${dy \over dx}sinx=y$ (* 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 {sinx \over {1 + 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 {{sin^2 x + cos^2 x} \over cos^2 x}}