Quiz — 6533eac0ba26b_exercice 3- equation differentielle.pdf

Question 1 sur 10 20:00

[{"id":30320,"question":"Quelle est la forme générale d'une équation différentielle linéaire du premier ordre ?","option_a":"y' + a(x)y = b(x)","option_b":"y'' + a(x)y' + b(x)y = c(x)","option_c":"y' = f(x,y)","option_d":"y = f(x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La forme générale d'une équation différentielle linéaire du premier ordre est y' + a(x)y = b(x), où a(x) et b(x) sont des fonctions continues.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"y' + a(x)y = b(x)\", \"b\": \"y'' + a(x)y' + b(x)y = c(x)\", \"c\": \"y' ","_debug_options_count":4},{"id":30321,"question":"L'équation différentielle y' = 2y est une équation à variables séparables.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai. On peut écrire dy\/y = 2dx, ce qui permet de séparer les variables x et y avant intégration.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":30322,"question":"Quelle est la solution générale de l'équation différentielle y' + 3y = 0 ?","option_a":"y = Ce^{-3x}","option_b":"y = Ce^{3x}","option_c":"y = Cx + D","option_d":"y = C sin(3x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La solution générale est y = Ce^{-3x}, obtenue en résolvant l'équation caractéristique r + 3 = 0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"y = Ce^{-3x}\", \"b\": \"y = Ce^{3x}\", \"c\": \"y = Cx + D\", \"d\": \"y = C","_debug_options_count":4},{"id":30323,"question":"L'équation différentielle y'' - 4y' + 4y = 0 est une équation du second ordre homogène.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai. C'est une équation linéaire du second ordre homogène (second membre nul).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":30324,"question":"Quelle méthode utilise-t-on pour résoudre une équation différentielle du second ordre à coefficients constants ?","option_a":"Résolution par séparation des variables","option_b":"Résolution de l'équation caractéristique","option_c":"Intégration directe","option_d":"Méthode de variation des constantes","option_e":"","option_f":"","bonne_reponse":"b","explication":"Pour une équation linéaire du second ordre à coefficients constants, on résout l'équation caractéristique associée.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Résolution par séparation des variables\", \"b\": \"Résolution de ","_debug_options_count":4},{"id":30325,"question":"L'équation différentielle y' = y^2 est une équation différentielle non linéaire.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai. L'équation n'est pas linéaire car le terme y^2 n'est pas de la forme linéaire en y.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":30326,"question":"Quelle est la solution générale de l'équation différentielle y'' + y = 0 ?","option_a":"y = Ce^x + De^{-x}","option_b":"y = C cos(x) + D sin(x)","option_c":"y = Cx + D","option_d":"y = Ce^{ix} + De^{-ix}","option_e":"","option_f":"","bonne_reponse":"b","explication":"La solution générale est y = C cos(x) + D sin(x), obtenue à partir des racines complexes de l'équation caractéristique.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"y = Ce^x + De^{-x}\", \"b\": \"y = C cos(x) + D sin(x)\", \"c\": \"y = Cx","_debug_options_count":4},{"id":30327,"question":"Une équation différentielle du premier ordre admet toujours une solution unique.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Faux. L'existence et l'unicité dépendent des conditions initiales et de la continuité des fonctions a(x) et b(x) (théorème de Cauchy-Lipschitz).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":30328,"question":"Quelle est la solution particulière de l'équation différentielle y' - 2y = e^{3x} ?","option_a":"y_p = Ae^{3x}","option_b":"y_p = Ae^{2x}","option_c":"y_p = Axe^{3x}","option_d":"y_p = A cos(3x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La solution particulière est de la forme y_p = Ae^{3x}, car 3 n'est pas une racine de l'équation caractéristique r - 2 = 0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"y_p = Ae^{3x}\", \"b\": \"y_p = Ae^{2x}\", \"c\": \"y_p = Axe^{3x}\", \"d\":","_debug_options_count":4},{"id":30329,"question":"L'équation différentielle y' = x + y est une équation différentielle linéaire.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Vrai. Elle peut s'écrire y' - y = x, qui est une équation linéaire du premier ordre.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4}]