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

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Line 4: Line 4:
 
<math>y''=0</math><br/>
 
<math>y''=0</math><br/>
 
D.h. y zweimal differenziert ist 0, da kann y maximal x hoch zwei sein (Polynom). Homogene Lösung (allgemein)<br/>
 
D.h. y zweimal differenziert ist 0, da kann y maximal x hoch zwei sein (Polynom). Homogene Lösung (allgemein)<br/>
−
<math>y_h=ax^2+bx+c</math><br/>
+
<math>y_h=ax+b</math><br/>
 
- spezielle Lösung -<br/>
 
- spezielle Lösung -<br/>
 
<math>y_{sp}''=-6x</math><br/>
 
<math>y_{sp}''=-6x</math><br/>
Line 13: Line 13:
 
- Gesamtlösung -<br/>
 
- Gesamtlösung -<br/>
 
<math>y=y_{sp}+y_h</math><br/>
 
<math>y=y_{sp}+y_h</math><br/>
−
<math>y=-x^3+ax^2+bx+c</math><br/>
+
<math>y=-x^3+ax+b</math><br/>
 
(a,b,c beliebig)
 
(a,b,c beliebig)
 
= 4.1 e) =
 
= 4.1 e) =
Line 19: Line 19:
 
- homogene -<br/>
 
- homogene -<br/>
 
<math>y_h''=0</math><br/>
 
<math>y_h''=0</math><br/>
−
<math>y_h=ax^2+bx+c</math><br/>
+
<math>y_h=ax+b</math><br/>
 
- spezielle -<br/>
 
- spezielle -<br/>
 
<math>y_{sp}''=-6x+3</math><br/>
 
<math>y_{sp}''=-6x+3</math><br/>
Line 26: Line 26:
 
- Gesamtlösung -<br/>
 
- Gesamtlösung -<br/>
 
<math>y=y_{sp}+y_h</math><br/>
 
<math>y=y_{sp}+y_h</math><br/>
−
<math>y=-x^3+ { 3 \over 2} x^2 +ax^2+bx+c</math><br/>
+
<math>y=-x^3+ { 3 \over 2} x^2 +ax+b</math><br/>
−
Da a,b,c beliebig - im speziellen a - ist die allgemeine Lösung (diesmal ein 'anderes' a):<br/>
 
−
<math>y=-x^3+ax^2+bx+c</math><br/>
 
 
= 4.1 f) =
 
= 4.1 f) =
 
<math>y''+12x^2-4x=1</math><br/>
 
<math>y''+12x^2-4x=1</math><br/>
 
- homogene (wie schon zwei Mal) -<br/>
 
- homogene (wie schon zwei Mal) -<br/>
−
<math>y_h=ax^2+bx+c</math><br/>
+
<math>y_h=ax+b</math><br/>
 
- spezeille -<br/>
 
- spezeille -<br/>
 
<math>y_{sp}''=-12x^2+4x+1</math><br/>
 
<math>y_{sp}''=-12x^2+4x+1</math><br/>
 
<math>y_{sp}'=-4x^3+2x^2+x</math><br/>
 
<math>y_{sp}'=-4x^3+2x^2+x</math><br/>
 
<math>y_{sp}=-x^4+{2 \over 3} x^3+ {1 \over 2} x^2</math><br/>
 
<math>y_{sp}=-x^4+{2 \over 3} x^3+ {1 \over 2} x^2</math><br/>
−
Wir wissen schon, dass uns wegen der homogenen Lösung der Faktor vor dem <math>x^2</math> wurscht ist<br/>
 
 
<math>y_{sp2}=-x^4+{2 \over 3} x^3</math><br/>
 
<math>y_{sp2}=-x^4+{2 \over 3} x^3</math><br/>
 
- Gesmatlösung - <br/>
 
- Gesmatlösung - <br/>
−
<math>y=-x^4+ {2 \over 3} x^3 + ax^2+bx+c</math>
+
<math>y=-x^4+ {2 \over 3} x^3 + + {1 \over 2} x^2 +ax+b</math>

Revision as of 08:28, 4 December 2008

4.1 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=0}
- homogene -
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}
D.h. y zweimal differenziert ist 0, da kann y maximal x hoch zwei sein (Polynom). Homogene Lösung (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_h=ax+b}
- spezielle Lösung -
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}
Einfach zweimal 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 y_{sp}'=-3x^2}
(kein +C, da man ja nur eine spezielle Lösung sucht!)
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}
- Gesamtlösung -
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_{sp}+y_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=-x^3+ax+b}
(a,b,c beliebig)

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

4.1 f)

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

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