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| Vorläufig aufgeschoben... <math>M_b</math> ? Biegung ? | | Vorläufig aufgeschoben... <math>M_b</math> ? Biegung ? |
| | | |
− | = -4.7 g) = | + | = - 4.7 g) = |
| <math>y'-y^2cosx=0</math><br/> | | <math>y'-y^2cosx=0</math><br/> |
| <math>y'=y^2cosx</math><br/> | | <math>y'=y^2cosx</math><br/> |
| <math>{dy \over dx} =y^2cosx</math> (* dx / <math>y^2</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^2}} dy = cosx dx</math> (integrieren)<br/> |
− | <math>{-1 \over y} = -sinx -c</math><br/> | + | <math>{-1 \over y} = sinx +c</math><br/> |
− | <math>y={1 \over {sinx + c}}</math><br/> | + | <math>y={-1 \over {sinx + c}}</math><br/> |
| Probe:<br/> | | Probe:<br/> |
| <math>y'={ cosx \over {(sinx +c)^2}}</math><br/> | | <math>y'={ cosx \over {(sinx +c)^2}}</math><br/> |
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| <math>y' cosx = -y sinx</math><br/> | | <math>y' cosx = -y sinx</math><br/> |
| <math>{dy \over dx} cosx = -y sinx</math> (* dx / cosx / y)<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>{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({-1 \over cosx})+c_1</math> (e^ <math>c_2=e^{c_1}</math>)<br/> | + | |
− | <math>y = c_2{-1 \over cosx}</math> (<math>c_3=-c_2</math>)<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_3 \over cosx}</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/> | | Probe:<br/> |
− | <math>y'=c_3 {sinx \over cosx}</math><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' cosx = c_3 sinx</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 sinx = c_3 {sin^2 x \over cosx}</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/> |
allgemein
homogene DGL
(oder auch höhere Ableitungen von y)
inhomogene DGL
Gelöst witd zuerst die homogene DGL - die Lösung der inhomogenen ist eine (irgend eine) Löung der inhomogenen plus die allgemeine Löung der homogenen
- 4.1 d)
- homogene -
D.h. y zweimal differenziert ist 0, da kann y maximal x sein (Polynom ersten Grades). Homogene Lösung (allgemein)
- spezielle Lösung -
Einfach zweimal integrieren:
(kein +C, da man ja nur eine spezielle Lösung sucht!)
- Gesamtlösung -
(a,b beliebig)
- 4.1 e)
- homogene -
- spezielle -
- Gesamtlösung -
- 4.1 f)
- homogene (wie schon zwei Mal) -
- spezeille -
- Gesmatlösung -
- 4.2 c)
- allgemein -
[1]
[2]
Jetzt in [2] laut Anfangsbedingung einsetzen:
Jetzt in [1] laut Anfangsbedingung einsetzen:
- 4.2 d)
- allgemein -
[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)