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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/> |
Line 159: |
Line 159: |
| <math>y' cosx + y sinx = c_3 (-sinx cosx + sinx cosx) = 0</math><br/> | | <math>y' cosx + y sinx = c_3 (-sinx cosx + sinx cosx) = 0</math><br/> |
| | | |
− | = -4.7 i) = | + | = - 4.7 i) = |
| <math>y'sinx + y=0</math><br/> | | <math>y'sinx + y=0</math><br/> |
| <math>y'sinx=-y</math><br/> | | <math>y'sinx=-y</math><br/> |
Line 172: |
Line 172: |
| <math>y' sinx = -y</math> passt<br/> | | <math>y' sinx = -y</math> passt<br/> |
| | | |
− | = -4.8 g) = | + | = - 4.8 g) = |
| <math>x y' + x y = y</math><br/> | | <math>x y' + x y = y</math><br/> |
| <math>y(1)=1</math><br/> | | <math>y(1)=1</math><br/> |
Line 201: |
Line 201: |
| <math>-{{ln(2y - 1)} \over 2} = 2 \sqrt{x} + c_1</math><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>-ln(2y - 1) = 4 \sqrt{x} + c_2</math> (e^ )<br/> |
− | <math>{1 \over {2y -1}} = c_3 e^{4 x^{1/2}}</math><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^{1/2}}}</math><br/> | + | <math>2y -1 = {c_4 \over e^{4 x^{1 \over 2}}}</math><br/> |
− | <math>2y = {c_4 \over e^{4 x^{1/2}}} + 1</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^{1/2}}}} + {1 \over 2}</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_4 \cdot 2 \cdot 4 \cdot {1 \over 2} x^{-1 \over 2} {e^{4 x^{1/2}}} } \over {({2 e^{4 x^{1/2}})}^2} }</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'={ {-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 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'={ {-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>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>={ -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/> | + | <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) = | | = - 6.8 h) = |
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)
![{\displaystyle \int {2 \over {\sqrt[{3}]{4x-1}}}\,dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5dfbf9a0390a799d0058227a77ee2b3b7359381)



![{\displaystyle \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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e7b0bb04d27bec316c6f4a816deded7f77cbf5b)


![{\displaystyle ={1 \over 3}{\sqrt[{3}]{(4x-1)^{2}}}+c}](https://wikimedia.org/api/rest_v1/media/math/render/svg/01ed54ce87d10301b4681f8ea1c13e43987d0f9f)
- 6.25 d)



