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temata:17a-matematicka_analyza:diferencialni_rovnice [2011/03/15 15:50] vagabund |
temata:17a-matematicka_analyza:diferencialni_rovnice [2011/04/21 11:46] (aktuální) vagabund |
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|---|---|---|---|
| Řádek 32: | Řádek 32: | ||
| **Actual Solution** - GS s dosazením IV | **Actual Solution** - GS s dosazením IV | ||
| - | **Implicitní/Explicitní řešení** - y = f(t)/0 = f(y,t) | + | **Explicitní/Implicitní řešení** - y = f(t)/0 = f(y,t) |
| </box> | </box> | ||
| Řádek 140: | Řádek 140: | ||
| <m>{2y - 4}{dy} = {3x^2 + 4x - 4}{dx}</m> | <m>{2y - 4}{dy} = {3x^2 + 4x - 4}{dx}</m> | ||
| - | <m>y^2 + 4y = x^3 + 2x^2 - 4x + k</m> | + | <m>y^2 - 4y = x^3 + 2x^2 - 4x + k</m> |
| partikularni reseni: | partikularni reseni: | ||
| - | <m>9 + 12 = 1 + 2 - 4 + k</m> | + | <m>9 - 12 = 1 + 2 - 4 + k</m> |
| - | <m>k = 22</m> | + | <m>k = -2</m> |
| - | <m>y^2 + 4y = x^3 + 2x^2 - 4x + 22</m> | + | <m>y^2 - 4y = x^3 + 2x^2 - 4x - 2</m> |
| </box> | </box> | ||
| Řádek 196: | Řádek 196: | ||
| | <m>asin(\beta t)</m> | <m>Acos(\beta t) + Bsin(\beta t)</m> | | | <m>asin(\beta t)</m> | <m>Acos(\beta t) + Bsin(\beta t)</m> | | ||
| | <m>acos(\beta t) + asin(\beta t)</m> | <m>Acos(\beta t) + Bsin(\beta t)</m> | | | <m>acos(\beta t) + asin(\beta t)</m> | <m>Acos(\beta t) + Bsin(\beta t)</m> | | ||
| + | | <m>3t^2 + 4t + 6</m> | <m>At^2 + Bt + C</m> | | ||
| Dopočítáme homogenní a dostaneme: <m>y(t) = Y_P(t) + Y_H(t)</m> | Dopočítáme homogenní a dostaneme: <m>y(t) = Y_P(t) + Y_H(t)</m> | ||
| + | |||
| + | </box> | ||
| + | |||
| + | == příklad == | ||
| + | <box round 90% green|**příklad**> | ||
| + | |||
| + | <m>y\prime\prime + 3y\prime - 10y = 0</m> | ||
| + | |||
| + | charakteristicka rovnice: | ||
| + | |||
| + | <m>{\lambda}^2 + 3{\lambda} - 10 = 0</m> | ||
| + | |||
| + | <m>({\lambda} + 5)({\lambda} - 2) = 0</m> | ||
| + | |||
| + | <m>{\lambda_1} = 2</m> | ||
| + | |||
| + | <m>{\lambda_2} = -5</m> | ||
| + | |||
| + | reseni: | ||
| + | |||
| + | <m>y = c_1e^{2t} + c_2e^{-5t}</m> | ||
| + | |||
| + | // | ||
| + | // | ||
| + | |||
| + | <m>y\prime\prime - 4y\prime + 4y = 0</m> | ||
| + | |||
| + | charakteristicka rovnice: | ||
| + | |||
| + | <m>{\lambda}^2 - 4{\lambda} + 4 = 0</m> | ||
| + | |||
| + | <m>{({\lambda} -2)}^2 = 0</m> | ||
| + | |||
| + | <m>{\lambda} = 2</m> | ||
| + | |||
| + | reseni: | ||
| + | |||
| + | <m>y = c_1e^{2t} + c_2te^{2t}</m> | ||
| + | |||
| + | // | ||
| + | // | ||
| + | |||
| + | <m>y\prime\prime - 4y\prime + 9y = 0</m> | ||
| + | |||
| + | charakteristicka rovnice: | ||
| + | |||
| + | <m>{\lambda}^2 - 4{\lambda} + 9 = 0</m> | ||
| + | |||
| + | <m>D = 16 - 36 = - 20</m> | ||
| + | |||
| + | <m>{\lambda}_{1,2} = {4 \pm 2i\sqrt{5}}/2</m> | ||
| + | |||
| + | <m>{\lambda}_{1,2} = 2 \pm i\sqrt{5}</m> | ||
| + | |||
| + | reseni: | ||
| + | |||
| + | <m>y = c_1e^{2t}sin(\sqrt{5}t) + c_2e^{2t}cos(\sqrt{5}t)</m> | ||
| + | </box> | ||
| + | |||
| + | <box round 90% green|**příklad**> | ||
| + | |||
| + | <m>y\prime\prime - 4y\prime - 12y = 3e^{5t}</m> | ||
| + | |||
| + | 1.) partikularni reseni hadanim | ||
| + | |||
| + | <m>Y_p = Ae^{5t}</m> | ||
| + | |||
| + | <m>Y_p\prime = 5Ae^{5t}</m> | ||
| + | |||
| + | <m>Y_p\prime\prime = 25Ae^{5t}</m> | ||
| + | |||
| + | <m>25Ae^{5t} - 20Ae^{5t} - 12Ae^{5t} = 3e^{5t}</m> | ||
| + | |||
| + | <m>-7Ae^{5t} = 3e^{5t}</m> | ||
| + | |||
| + | <m>A = -3/7</m> | ||
| + | |||
| + | <m>Y_p = -3/7e^{5t}</m> | ||
| + | |||
| + | 2.) homogenní řešení | ||
| + | |||
| + | <m>{\lambda}^2 - 4{\lambda} - 12 = 0</m> | ||
| + | |||
| + | <m>(\lambda - 6)(\lambda + 2) = 0</m> | ||
| + | |||
| + | <m>\lambda_1 = 6</m> | ||
| + | |||
| + | <m>\lambda_2 = -2</m> | ||
| + | |||
| + | <m>Y_h = c_1e^{6t} + c_2e(-2t)</m> | ||
| + | |||
| + | 3.) <m>Y = Y_h + Y_p</m> | ||
| + | |||
| + | <m>Y = c_1e^{6t} + c_2e(-2t) + -3/7e^{5t}</m> | ||
| + | |||
| </box> | </box> | ||
| Řádek 203: | Řádek 299: | ||
| ===== Potvrzení ===== | ===== Potvrzení ===== | ||
| - | <doodle single login|XX> | + | <doodle single login|17a3> |
| ^ OK ^ !!! ^ | ^ OK ^ !!! ^ | ||
| </doodle> | </doodle> | ||
