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

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−
= 4.1 d) =
+
= allgemein =
 +
homogene DGL<br/>
 +
<math>ay''+by'+cy=0</math> (oder auch höhere Ableitungen von y)<br/>
 +
inhomogene DGL<br/>
 +
<math>ay''+by'+cy=f(x)</math><br/>
 +
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) =
 
<math>y''+6x=0</math><br/>
 
<math>y''+6x=0</math><br/>
 
- homogene -<br/>
 
- homogene -<br/>
 
<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 sein (Polynom ersten Grades). 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 20:
 
- 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 beliebig)
−
= 4.1 e) =
+
= - 4.1 e) =
 
<math>y''+6x-3=0</math><br/>
 
<math>y''+6x-3=0</math><br/>
 
- 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/>
 
<math>y_{sp}'=-3x^2+3x</math><br/>
 
<math>y_{sp}'=-3x^2+3x</math><br/>
−
<math>y_{sp}=-x^3+3/2 x^2</math><br/>
+
<math>y_{sp}=-x^3+ { 3 \over 2} x^2</math><br/>
 
- Gesamtlösung -<br/>
 
- Gesamtlösung -<br/>
 
<math>y=y_{sp}+y_h</math><br/>
 
<math>y=y_{sp}+y_h</math><br/>
−
<math>y=-x^3+3/2 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/>
+
= - 4.1 f) =
−
<math>y=-x^3+ax^2+bx+c</math><br/>
 
−
= 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+2/3 x^3+ 1/2 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 x^2 wurscht ist<br/>
+
<math>y_{sp2}=-x^4+{2 \over 3} x^3</math><br/>
−
<math>y_{sp2}=-x^4+2/3 x^3<math><br/>
 
 
- Gesmatlösung - <br/>
 
- Gesmatlösung - <br/>
−
<math>y=-x^4+2/3 x^3 + ax^2+bx+c</math>
+
<math>y=-x^4+ {2 \over 3} x^3 + {1 \over 2} x^2 +ax+b</math>
 +
 
 +
= - 4.2 c) =
 +
<math>y''-x+1=0</math><br/>
 +
<math>y(1)=0</math><br/>
 +
<math>y'(1)=0</math><br/>
 +
- allgemein -<br/>
 +
<math>y''=x-1</math><br/>
 +
<math>y_h''=0</math><br/>
 +
<math>y_h=ax+b</math><br/>
 +
<math>y_{sp}''=x-1</math><br/>
 +
<math>y_{sp}'={x^2 \over 2} -x</math><br/>
 +
<math>y_{sp}={x^3 \over 6} - {x^2 \over 2}</math><br/>
 +
<math>y={x^3 \over 6} - {x^2 \over 2} + ax + b</math> [1]<br/>
 +
<math>y'={x^2 \over 2} - x + a</math> [2] <br/>
 +
Jetzt in [2] laut Anfangsbedingung einsetzen:<br/>
 +
<math>0={1 \over 2} - 1 + a</math><br/>
 +
<math>a={1 \over 2}</math><br/>
 +
Jetzt in [1] laut Anfangsbedingung einsetzen:<br/>
 +
<math>0={1 \over 6} - {1 \over 2} + a + b</math><br/>
 +
<math>0={1 \over 6} - {1 \over 2} + {1 \over 2} + b</math><br/>
 +
<math>b=-{1 \over 6}</math><br/>
 +
<math>y={x^3 \over 6} - {x^2 \over 2} + {1 \over 2} x -{1 \over 6}</math><br/>
 +
 
 +
= - 4.2 d) =
 +
<math>y''+6x^2=1-x</math><br/>
 +
<math>y(0)=2</math><br/>
 +
<math>y'(0)=5</math><br/>
 +
- allgemein -<br/>
 +
<math>y''=-6x^2-x+1</math><br/>
 +
<math>y_h=ax+b</math><br/>
 +
<math>y_{sp}''=-6x^2-x+1</math><br/>
 +
<math>y_{sp}'=-2x^3-{1 \over 2} x^2 + x</math><br/>
 +
<math>y_{sp}=-{1 \over 2} x^4 -{1 \over 6} x^3 + {1 \over 2} x^2</math><br/>
 +
<math>y=-{1 \over 2} x^4 -{1 \over 6} x^3 + {1 \over 2} x^2+ax+b</math> [1]<br/>
 +
<math>y'=-2x^3-{1 \over 2} x^2+x+a</math> [2]<br/>
 +
Einsetzen in [2]<br/>
 +
<math>5=0+0+0+a</math><br/>
 +
<math>a=5</math><br/>
 +
Einsetzen in [1]<br/>
 +
<math>2=0+0+0+0+b</math><br/>
 +
<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) =
 +
<math>y''=2x-1</math><br/>
 +
<math>y(0)=1</math><br/>
 +
<math>y(6)=1</math><br/>
 +
<math>y_h=ax+b</math><br/>
 +
<math>y_{sp}''=2x-1</math><br/>
 +
<math>y_{sp}'=x^2-x</math><br/>
 +
<math>y_{sp}={1 \over 3} x^3 - {1 \over 2} x^2</math><br/>
 +
<math>y={1 \over 3} x^3 - {1 \over 2} x^2+ax+b</math><br/>
 +
Einsetzen y(0)=1<br/>
 +
<math>1=0 - 0+0+b</math><br/>
 +
<math>b=1</math><br/>
 +
Einsetzen y6)=1<br/>
 +
<math>1={1 \over 3} 6^3 - {1 \over 2} 6^2+a6+b</math><br/>
 +
<math>1=72-18+6a+1</math><br/>
 +
<math>-54=6a</math><br/>
 +
<math>a=9</math><br>
 +
<math>y={1 \over 3} x^3 - {1 \over 2} x^2+9x+1</math><br/>
 +
 
 +
= - 4.3 b) =
 +
<math>y''-x+3=0</math><br/>
 +
<math>y(3)=1</math><br/>
 +
<math>y(9)=10</math><br/>
 +
<math>y_h=ax+b</math><br/>
 +
<math>y_{sp}''=x-3</math><br>
 +
<math>y_{sp}'={1 \over 2} x^2 - 3x</math><br/>
 +
<math>y_{sp}={1 \over 6} x^3 - {3 \over 2} x^2</math><br/>
 +
<math>y={1 \over 6} x^3 - {3 \over 2} x^2+ax+b</math><br/>
 +
Einsetzen => lineares Gleichungssystem:<br/>
 +
<math>1={1 \over 6} 3^3 - {3 \over 2} 3^2+a3+b</math><br/>
 +
<math>1={27 \over 6} - {27 \over 2}+ 3a+b</math><br/>
 +
<math>3a+b={6 \over 6} - {27 \over 6} + {81 \over 6}={60 \over  6}=10</math> [1]<br/>
 +
<math>1={1 \over 6} 9^3 - {3 \over 2} 9^2+a9+b</math><br/>
 +
<math>1={729 \over 6} - {243 \over 2} + 9a+b</math><br/>
 +
<math>9a+b={6 \over 6} - {729 \over 6} + {729 \over 6}=1</math> [2]<br>
 +
Subtrahiere [1] von [2]<br/>
 +
<math>6a=1-10=-9</math><br/>
 +
<math>a=-{9 \over 6}=-{3 \over 2}</math><br/>
 +
Einsetzen in [1]<br/>
 +
<math>3(-{3 \over 2})+b=10</math><br/>
 +
<math>b=10+{9 \over 2}={29 \over 2}</math><br/>
 +
<math>y={1 \over 6} x^3 - {3 \over 2} x^2-{3 \over 2}x+{29 \over 2}</math><br/>
 +
 
 +
= - 4.4 =
 +
Vorläufig aufgeschoben... <math>M_b</math> ? Biegung ?
 +
 
 +
= - 4.7 g) =
 +
<math>y'-y^2cosx=0</math><br/>
 +
<math>y'=y^2cosx</math><br/>
 +
<math>{dy \over dx} =y^2cosx</math> (* dx / <math>y^2</math>)<br/>
 +
<math>{1 \over {y^2}} dy = cosx dx</math> (integrieren)<br/>
 +
<math>{-1 \over y} = sinx +c</math><br/>
 +
<math>y={-1 \over {sinx + c}}</math><br/>
 +
Probe:<br/>
 +
<math>y'={ cosx \over {(sinx +c)^2}}</math><br/>
 +
<math>y^2={1 \over {(sinx +c)^2}}</math><br/>
 +
<math>y'=y^2 cosx</math> passt<br/>
 +
 
 +
= - 4.7 h) =
 +
<math>y' cosx + y sinx = 0</math><br/>
 +
<math>y' cosx = -y sinx</math><br/>
 +
<math>{dy \over dx} cosx = -y sinx</math> (* dx / cosx / y)<br/>
 +
<math>{1 \over y} dy = { -sinx \over cosx } dx</math> (integrieren)<br/>
 +
 
 +
<math>lny=-ln({2 \over {1 + cosx}}) + ln(-{{2 cosx} \over {1 + cosx}}) +c_1=ln(cosx)+c_1</math> (e^ <math>c_2=e^{c_1}</math>)<br/>
 +
<math>y = c_2 cosx</math><br/>
 +
Probe:<br/>
 +
<math>y'=c_2 -sinx</math><br/>
 +
<math>y' cosx = c_2 -sinx cosx</math><br/>
 +
<math>y sinx = c_2 sinx cosx</math><br/>
 +
<math>y' cosx + y sinx = c_3 (-sinx cosx + sinx cosx) = 0</math><br/>
 +
 
 +
= - 4.7 i) =
 +
<math>y'sinx + y=0</math><br/>
 +
<math>y'sinx=-y</math><br/>
 +
<math>{dy \over dx} sinx = -y</math> (* dx / y / sinx)<br/>
 +
<math>{1 \over y} dy = {-1 \over sinx } dx</math> (integrieren)<br/>
 +
<math>lny = -ln({sinx \over {1 + cosx}}) + c_1</math> (e^ <math>c_2=e^{c_1}</math>)<br/>
 +
<math>y=c_2 {{1 + cosx} \over sinx}</math><br/>
 +
Probe:<br/>
 +
<math>y'=c_2 {{-sinx sinx - (1+cosx) cosx} \over {sin^2 x}}</math><br/>
 +
<math>y'=c_2{{-sin^2 x -cosx -cos^2 x} \over {sin^2 x}}</math> (<math>sin^2x+cos^2x = 1</math>)<br/>
 +
<math>y'=c_2{{-1 -cosx} \over {sin^2 x}}</math><br/>
 +
<math>y' sinx = -y</math> passt<br/>
 +
 
 +
= - 4.8 g) =
 +
<math>x y' + x y = y</math><br/>
 +
<math>y(1)=1</math><br/>
 +
<math>x y' = y (1-x)</math><br/>
 +
<math>x {dy \over dx} = y (1-x)</math> (* dx / x / y)<br/>
 +
<math>{1 \over y} dy = {{1-x} \over x} dx</math> (integrieren)<br/>
 +
<math>lny = lnx -x +c_1</math><br/>
 +
<math>y=c_2 {x \over {e^x}}</math><br/>
 +
Probe:<br/>
 +
<math>y'=c_2{{1 e^x - x e^x \over (e^x)^2}}</math><br/>
 +
<math>y'=c_2{{e^x (1-x)} \over {e^x e^x}}</math><br/>
 +
<math>y'=c_2{(1-x) \over e^x}</math><br/>
 +
<math>x y'= c_2{{x(1-x)} \over e^x}</math><br/>
 +
<math>x y = c_2 {x^2 \over e^x}</math><br/>
 +
<math>x y' + x y = c_2 {{x(1-x) + x^2} \over e^x}</math><br/>
 +
<math>x y' + x y = c_2 {{x -x^2 + x^2} \over e^x}</math><br/>
 +
<math>x y' + x y = c_2 {x \over e^x}</math> passt<br/>
 +
Ensetzen y(1)=1<br/>
 +
<math>1=c_2 {1 \over {e^1}}</math><br/>
 +
<math>c_2=e</math><br/>
 +
<math>y=e{x \over {e^x}}</math>
 +
 
 +
= - 4.8 h) =
 +
<math>y' \sqrt{x} + 2y=1</math><br/>
 +
<math>y(0)=1</math><br/>
 +
<math>{dy \over dx} \sqrt{x} = 1 - 2y</math> (* dx / sqrt(x) / (1-2y))<br/>
 +
<math>{ 1 \over {1-2y}} dy = { 1 \over \sqrt{x}} dx = x^{-{1 \over 2}} dx</math> (integrieren)<br/>
 +
<math>-{{ln(2y - 1)} \over 2} = 2 \sqrt{x} + c_1</math><br/>
 +
<math>-ln(2y - 1) = 4 \sqrt{x} + c_2</math> (e^ )<br/>
 +
<math>{1 \over {2y -1}} = c_3 e^{4 x^{1 \over 2}}</math><br/>
 +
<math>2y -1 = {c_4 \over e^{4 x^{1 \over 2}}}</math><br/>
 +
<math>2y = {c_4 \over e^{4 x^{1 \over 2}}} + 1</math><br/>
 +
<math>y = {c_4 \over {2 e^{4 x^{1 \over 2}}}} + {1 \over 2}</math><br/>
 +
Probe:<br/>
 +
<math>y'={ {-c_4 \cdot 2 \cdot 4 \cdot {1 \over 2} x^{-1 \over 2} {e^{4 x^{1 \over 2}}} } \over {({2 e^{4 x^{1/2}})}^2} }</math><br/>
 +
<math>y'={ {-c_4 \cdot 4 \cdot {1 \over 2} x^{-1 \over 2}  } \over {2 e^{4 x^{1/2}}} }</math><br/>
 +
<math>y'={ {-c_4 \cdot x^{-1 \over 2}  } \over {e^{4 x^{1/2}}} }</math><br/>
 +
<math>y' \sqrt{x}={ {-c_4 \cdot x^{-1 \over 2} \cdot x^{1 \over 2} } \over {e^{4 x^{1 \over 2}}} }</math><br/>
 +
<math>={ -c_4 \over {e^{4 x^{1 \over 2}}} }</math><br/>
 +
<math>y' \sqrt{x} + 2y={ {-c_4 \over {e^{4 x^{1 \over 2}}}} + {c_4 \over e^{4 x^{1/2}}} + 1}=1</math> passt<br/>
 +
 
 +
= - 6.8 h) =
 +
<math>\int {1 \over {sin^2 x}} \, dx</math><br/>
 +
<math>({u \over v})'={{u'v - u v'} \over {v^2}}</math><br/>
 +
<math>v=sinx</math><br/>
 +
<math>u'v - u v'= 1</math><br/>
 +
<math>v^2=sin^2 x</math><br/>
 +
<math>v'=cosx</math><br/>
 +
<math>u' sinx - u cosx = 1</math> (das klappt mit u'=sinx und u=-cosx)<br/>
 +
<math>u=-cosx</math><br/>
 +
<math>\int {1 \over {sin^2 x}} \, dx  = {-cosx \over sinx} +c = -{cosx \over sinx} +c</math><br/>
 +
 
 +
= - 6.8 i) =
 +
<math>\int {1 \over {1+x^2}}\, dx</math><br/>
 +
?
 +
 
 +
= - 6.9 e) =
 +
<math>f(x)={1 \over x}</math><br/>
 +
<math>P(e/4)</math><br/>
 +
<math>F(e)=4</math><br/>
 +
<math>F(x)=lnx + c</math><br/>
 +
<math>lne+c=4</math><br/>
 +
<math>1+c=4</math><br/>
 +
<math>c=3</math><br/>
 +
<math>F(x)=lnx+3</math><br/>
 +
 
 +
= - 6.9 f) =
 +
<math>f(x)=2^x</math><br/>
 +
<math>P(1/3)</math><br/>
 +
<math>F(1)=3</math><br/>
 +
<math>f(x)=e^{x ln(2)}=</math><br/>
 +
<math>F(x)={1 \over ln2}e^{x ln(2)}+c</math><br/>
 +
<math>3={1 \over ln2}e^{1 ln(2)}+c</math><br/>
 +
<math>3={2 \over ln2}+c</math><br/>
 +
<math>c=3-{2 \over ln2}</math><br/>
 +
<math>F(x)={1 \over ln2}e^{x ln(2)}+3-{2 \over ln2}</math><br/>
 +
 
 +
= - 6.22 d) =
 +
<math>\int { 1 \over \sqrt {1-x}} \, dx</math><br/>
 +
<math>u=1-x</math><br/>
 +
<math>{du \over dx}=-1</math><br/>
 +
<math>dx=-dx</math><br/>
 +
<math>\int { 1 \over \sqrt {1-x}} \, dx= -\int {1 \over \sqrt {u}}\, du=-\int {u^{-{1 \over 2}}}\, du</math><br/>
 +
<math>=-{1 \over 2} u^{1\over2}+c</math><br/>
 +
<math>=-{1 \over 2} (1-x)^{1\over2}+c</math><br/>
 +
<math>=-{1 \over 2} \sqrt{1-x}+c</math><br/>
 +
<math>={- \sqrt{1-x} \over 2}+c</math><br/>
 +
 
 +
= - 6.23 d) =
 +
<math>\int { 2a \over  {a+2x}} \, dx</math><br/>
 +
<math>u=a+2x</math><br/>
 +
<math>{du \over dx}=2</math><br/>
 +
<math>dx={du \over 2}</math><br/>
 +
<math>\int { 2a \over  {a+2x}} \, dx={1 \over 2} \int {2a \over u} \, du</math><br/>
 +
<math>={2a \over 2} lnu+c=a \cdot lnu+c</math><br/>
 +
<math>=a \cdot ln(a+2x)+c</math><br/>
 +
<math>=ln((a+2x)^a)+c</math><br/>
 +
 
 +
= - 6.24 d) =
 +
<math>\int { 2 \over \sqrt[3] {4x-1}} \, dx</math><br/>
 +
<math>u=4x-1</math><br/>
 +
<math>{du \over dx} = 4</math><br/>
 +
<math>dx = {du \over 4}</math><br/>
 +
<math>\int { 2 \over \sqrt[3] {4x-1}} \, dx = {1 \over 4} \int {2 \over \sqrt[3]{u}} \, du={2 \over 4} \int {u^{-{1 \over 3}}} \, du={1 \over 2} \int {u^{-{1 \over 3}}} \, du</math><br/>
 +
<math>={1 \over 2} \cdot {2 \over 3} u^{2 \over 3}+c</math><br/>
 +
<math>={2 \over 6} (4x-1)^{2 \over 3}+c</math><br/>
 +
<math>={1 \over 3} \sqrt[3]{(4x-1)^2}+c</math><br/>
 +
 
 +
= - 6.25 d) =
 +
<math>\int (e^{3x} - e^{-3x}) \, dx</math><br/>
 +
<math>=\int {e^{3x}} \,dx - \int {e^{-3x}} \, dx </math><br/>
 +
<math>={1 \over 3} e^{3x} - {-1 \over 3} e^{-3x}+c</math><br/>
 +
<math>={{e^{3x} + e^{-3x}} \over 3}+c</math><br/>

Latest revision as of 19:46, 4 December 2008

allgemein

homogene DGL
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 ay''+by'+cy=0} (oder auch höhere Ableitungen von y)
inhomogene DGL
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 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
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}

- 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 sein (Polynom ersten Grades). 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 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}

- 4.2 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''-x+1=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(1)=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'(1)=0}
- 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''=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''=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}
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-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 \over 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^3 \over 6} - {x^2 \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={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}
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} + {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}
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)=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}

[1]
[2]
Einsetzen in [2]


Einsetzen in [1]



- 4.3 a)









Einsetzen y(0)=1


Einsetzen y6)=1





- 4.3 b)









Einsetzen => lineares Gleichungssystem:


[1]


[2]
Subtrahiere [1] von [2]


Einsetzen in [1]



- 4.4

Vorläufig aufgeschoben...  ? Biegung ?

- 4.7 g)



(* dx / )
(integrieren)


Probe:


passt

- 4.7 h)



(* dx / cosx / y)
(integrieren)

(e^ )

Probe:




- 4.7 i)



(* dx / y / sinx)
(integrieren)
(e^ )

Probe:

()

passt

- 4.8 g)




(* dx / x / y)
(integrieren)


Probe:







passt
Ensetzen y(1)=1


- 4.8 h)



(* dx / sqrt(x) / (1-2y))
(integrieren)

(e^ )




Probe:





passt

- 6.8 h)







(das klappt mit u'=sinx und u=-cosx)


- 6.8 i)


?

- 6.9 e)









- 6.9 f)










- 6.22 d)










- 6.23 d)









- 6.24 d)









- 6.25 d)





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