Quiz — ~$Equations diff.docx

Question 1 sur 10 20:00

[{"id":72495,"question":"Quelle est la forme générale d’une équation différentielle du premier ordre ?","option_a":"y' + P(x)y = Q(x)","option_b":"y'' + P(x)y' = Q(x)","option_c":"y' = f(x,y)","option_d":"y = f(x) + C","option_e":"","option_f":"","bonne_reponse":"a","explication":"La forme générale d’une équation différentielle du premier ordre est y' + P(x)y = Q(x), où P(x) et Q(x) sont des fonctions de 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\": \"y' + P(x)y = Q(x)\", \"b\": \"y'' + P(x)y' = Q(x)\", \"c\": \"y' = f(x,y)","_debug_options_count":4},{"id":72496,"question":"Une équation différentielle est dite linéaire si elle peut s’écrire sous la forme y' + a(x)y = b(x).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C’est exact. Une équation différentielle linéaire du premier ordre a cette forme spécifique.","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":72497,"question":"Quelle méthode utilise-t-on pour résoudre une équation différentielle à variables séparables ?","option_a":"Intégration directe","option_b":"Séparation des variables puis intégration","option_c":"Substitution trigonométrique","option_d":"Méthode de variation de la constante","option_e":"","option_f":"","bonne_reponse":"b","explication":"La méthode consiste à séparer les variables x et y avant d’intégrer chaque côté 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\": \"Intégration directe\", \"b\": \"Séparation des variables puis inté","_debug_options_count":4},{"id":72498,"question":"Si y' = 2y, quelle est la solution générale de cette équation ?","option_a":"y = Ce^{2x}","option_b":"y = Cx^2","option_c":"y = C + 2x","option_d":"y = Ce^{-2x}","option_e":"","option_f":"","bonne_reponse":"a","explication":"La solution générale est y = Ce^{2x}, car en dérivant, on retrouve y' = 2Ce^{2x} = 2y.","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^{2x}\", \"b\": \"y = Cx^2\", \"c\": \"y = C + 2x\", \"d\": \"y = Ce^{-","_debug_options_count":4},{"id":72499,"question":"Les conditions initiales permettent de déterminer la constante d’intégration C.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Oui, les conditions initiales (comme y(x0) = y0) permettent de trouver la valeur spécifique de C.","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":72500,"question":"Quelle est la solution générale de l’équation différentielle y' + 3y = 0 ?","option_a":"y = Ce^{-3x}","option_b":"y = Cx^3","option_c":"y = C + 3x","option_d":"y = Ce^{3x}","option_e":"","option_f":"","bonne_reponse":"a","explication":"La solution générale est y = Ce^{-3x}, car y' = -3Ce^{-3x} = -3y, ce qui vérifie l’équation.","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 = Cx^3\", \"c\": \"y = C + 3x\", \"d\": \"y = Ce^{","_debug_options_count":4},{"id":72501,"question":"Pour résoudre une équation différentielle du premier ordre, il faut toujours commencer par vérifier si elle est à variables séparables.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Faux. Certaines équations nécessitent d’autres méthodes comme les facteurs intégrants ou la variation de la constante.","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":72502,"question":"Quelle est la solution de l’équation différentielle y' = y + x avec la condition initiale y(0) = 1 ?","option_a":"y = Ce^x - x - 1","option_b":"y = Ce^x + x + 1","option_c":"y = Ce^x - x + 1","option_d":"y = Ce^x + x - 1","option_e":"","option_f":"","bonne_reponse":"c","explication":"La solution générale est y = Ce^x - x - 1. Avec y(0) = 1, on trouve C = 2, donc y = 2e^x - x - 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"y = Ce^x - x - 1\", \"b\": \"y = Ce^x + x + 1\", \"c\": \"y = Ce^x - x + ","_debug_options_count":4},{"id":72503,"question":"Une équation différentielle non homogène a toujours une solution unique.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Faux. Une équation non homogène a une solution générale composée de la solution homogène + une solution particulière.","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":72504,"question":"Quelle méthode est adaptée pour résoudre y' + 2xy = x^2 ?","option_a":"Variables séparables","option_b":"Facteur intégrant","option_c":"Équation exacte","option_d":"Méthode de variation de la constante","option_e":"","option_f":"","bonne_reponse":"b","explication":"Cette équation est linéaire du premier ordre. On utilise un facteur intégrant μ(x) = e^{∫2x dx} = e^{x^2} pour la résoudre.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Variables séparables\", \"b\": \"Facteur intégrant\", \"c\": \"Équatio","_debug_options_count":4}]