Quiz — corr-eq-diff.pdf

Question 1 sur 10 20:00

[{"id":2017,"question":"Quelle est la forme générale d'une équation différentielle linéaire du premier ordre ?","option_a":"A. y' + a(x)y = b(x)","option_b":"B. y'' + a(x)y' = b(x)","option_c":"C. y' = a(x)y","option_d":"D. y = a(x)y' + b(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).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"A. y' + a(x)y = b(x)\", \"b\": \"B. y'' + a(x)y' = b(x)\", \"c\": \"C. y'","_debug_options_count":4},{"id":2018,"question":"L'équation différentielle y' = 2y est-elle linéaire ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Cette équation est linéaire car elle peut s'écrire sous la forme y' - 2y = 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":2019,"question":"Quelle méthode utilise-t-on pour résoudre une équation différentielle à variables séparables ?","option_a":"A. Intégration directe","option_b":"B. Séparation des variables et intégration","option_c":"C. Méthode des facteurs intégrants","option_d":"D. Résolution par substitution","option_e":"","option_f":"","bonne_reponse":"b","explication":"La méthode des variables séparables consiste à séparer les variables x et y avant d'intégrer les deux côtés de l'équation.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"A. Intégration directe\", \"b\": \"B. Séparation des variables et i","_debug_options_count":4},{"id":2020,"question":"Si y' + 3y = 0, quelle est la solution générale ?","option_a":"A. y = Ce^{-3x}","option_b":"B. y = Ce^{3x}","option_c":"C. y = Cx^3","option_d":"D. y = C\/x","option_e":"","option_f":"","bonne_reponse":"a","explication":"La solution générale est y = Ce^{-3x}, car l'équation est de la forme y' + a(x)y = 0 avec a(x) = 3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"A. y = Ce^{-3x}\", \"b\": \"B. y = Ce^{3x}\", \"c\": \"C. y = Cx^3\", \"d\":","_debug_options_count":4},{"id":2021,"question":"L'équation différentielle y' = y^2 est-elle linéaire ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Cette équation n'est pas linéaire car elle contient y^2, ce qui la rend non linéaire.","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":2022,"question":"Quelle est la solution générale de l'équation différentielle y' = 5 ?","option_a":"A. y = 5x + C","option_b":"B. y = Cx + 5","option_c":"C. y = 5x^2 + C","option_d":"D. y = C","option_e":"","option_f":"","bonne_reponse":"a","explication":"La solution générale est y = 5x + C, car l'intégrale de 5 est 5x + constante.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"A. y = 5x + C\", \"b\": \"B. y = Cx + 5\", \"c\": \"C. y = 5x^2 + C\", \"d\"","_debug_options_count":4},{"id":2023,"question":"Pour résoudre y' + 2xy = x, quelle méthode utilise-t-on ?","option_a":"A. Variables séparables","option_b":"B. Facteur intégrant","option_c":"C. Équation homogène","option_d":"D. Équation de Bernoulli","option_e":"","option_f":"","bonne_reponse":"b","explication":"On utilise un facteur intégrant ici, car l'équation est linéaire et de la forme y' + a(x)y = b(x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"A. Variables séparables\", \"b\": \"B. Facteur intégrant\", \"c\": \"C.","_debug_options_count":4},{"id":2024,"question":"L'équation différentielle y' = sin(x)y est-elle à variables séparables ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Oui, car on peut écrire dy\/y = sin(x)dx, ce qui permet de séparer les variables.","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":2025,"question":"Quelle est la solution de l'équation différentielle y' = 3y avec y(0) = 2 ?","option_a":"A. y = 2e^{3x}","option_b":"B. y = 3e^{2x}","option_c":"C. y = 2e^{-3x}","option_d":"D. y = 3x + 2","option_e":"","option_f":"","bonne_reponse":"a","explication":"La solution est y = 2e^{3x}, car y(0) = 2 impose C = 2 dans la solution générale y = Ce^{3x}.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"A. y = 2e^{3x}\", \"b\": \"B. y = 3e^{2x}\", \"c\": \"C. y = 2e^{-3x}\", \"","_debug_options_count":4},{"id":2026,"question":"L'équation différentielle y' + y = e^x est-elle homogène ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Non, car elle contient un terme non homogène (e^x). Une équation homogène ne contient que des termes en y et ses dérivées.","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}]