mat235 - MAT-235 Examen final de pratique AVERTISSEMENT...

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MAT-235 Examen final de pratique gp 3 dØcembre 2001 1 AVERTISSEMENT : voici un exemple d°examen final basØ sur celui de la session ±-98. Il est distribuØ ² titre indicatif. L°examen que vous aurez pourra OEtre plus long ou plus court, plus difficile ou facile que celui-ci. Il est possible que certains sujets traitØs ici ne le soient pas dans l’examen que vous aurez. ³ l’inverse, certains sujets non-traitØs ici pourraient se retrouver dans votre examen final. 1- [15 points] Calculez les transformØes de Laplace (directes ou inverses selon le cas) des fonctions suivantes : a) f t t t u t ( ) cos( ) ( ) = 2 2 c h b) f t t u t ( ) sin ( ) = a f 2 π c) F s e s s s ( ) = + + 3 2 8 22 d) F s s s s ( ) ( ) = + 1 4 2 c h . 2- [15 points] RØsolvez chacune des Øquations diffØrentielles suivantes en utilisant la transformØe de Laplace : a) y y tu t y y ″+ = = = ( ) ( ) ( ) 2 0 0 0 0 , , ; b) y y y e t y y t ″+ ′+ = = = 2 1 0 2 0 0 + , , δ ( ) ( ) ( ) . 3- [20 points] ConsidØrons l°Øquation diffØrentielle linØaire du deuxiLme ordre, xy y y y y ″+ ′− = = = 2 3 0 1 0 1 1 , , ( ) ( ) .
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