Revision as of 09:55, 4 December 2008

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]
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'={x^2 \over 2} - x + a} [2]
Jetzt in [2] laut Anfangsbedingung einsetzen:
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 0={1 \over 2} - 1 + a}
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={1 \over 2}}
Jetzt in [1] laut Anfangsbedingung einsetzen:
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 0={1 \over 6} - {1 \over 2} + a + b}
$0={1 \over 6}-{1 \over 2}+{1 \over 2}+b$
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 b=-{1 \over 6}}
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={x^3 \over 6} - {x^2 \over 2} + {1 \over 2} x -{1 \over 6}}

- 4.2 d)

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

- 4.3 a)

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

- 4.3 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''-x+3=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(3)=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(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}}

@@ Line 6: / Line 6: @@
 Gelöst witd zuerst die homogene DGL - die Lösung der inhomogenen ist eine (irgend eine) Löung der inhomogenen plus die allgemeine Löung der homogenen<br/>
 <math>y=y_{sp}+y_h</math>
-= 4.1 d) =
+= - 4.1 d) =
 <math>y''+6x=0</math><br/>
 - homogene -<br/>
@@ Line 22: / Line 22: @@
 <math>y=-x^3+ax+b</math><br/>
 (a,b beliebig)
-= 4.1 e) =
+= - 4.1 e) =
 <math>y''+6x-3=0</math><br/>
 - homogene -<br/>
@@ Line 34: / Line 34: @@
 <math>y=y_{sp}+y_h</math><br/>
 <math>y=-x^3+ { 3 \over 2} x^2 +ax+b</math><br/>
-= 4.1 f) =
+= - 4.1 f) =
 <math>y''+12x^2-4x=1</math><br/>
 - homogene (wie schon zwei Mal) -<br/>
@@ Line 46: / Line 46: @@
 <math>y=-x^4+ {2 \over 3} x^3 + {1 \over 2} x^2 +ax+b</math>
-= 4.2 c) =
+= - 4.2 c) =
 <math>y''-x+1=0</math><br/>
 <math>y(1)=0</math><br/>
@@ Line 68: / Line 68: @@
 <math>y={x^3 \over 6} - {x^2 \over 2} + {1 \over 2} x -{1 \over 6}</math><br/>
-= 4.2 d) =
+= - 4.2 d) =
 <math>y''+6x^2=1-x</math><br/>
 <math>y(0)=2</math><br/>
@@ Line 87: / Line 87: @@
 <math>b=2</math><br/>
 <math>y=-{1 \over 2} x^4 -{1 \over 6} x^3 + {1 \over 2} x^2+5x+2</math><br/>
-= 4.3 a) =
+= - 4.3 a) =
 <math>y''=2x-1</math><br/>
 <math>y(0)=1</math><br/>
@@ Line 106: / Line 106: @@
 <math>y={1 \over 3} x^3 - {1 \over 2} x^2+9x+1</math><br/>
-= 4.3 b) =
+= - 4.3 b) =
 <math>y''-x+3=0</math>
 <math>y(3)=1</math><br/>

Difference between revisions of "Jan Math 2008-12-05"