Quiz — Lycée Pilote Sfax - Equation Différentielle 1.pdf

Question 1 sur 10 20:00

[{"id":35590,"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,y)","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 données.","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":35591,"question":"La solution générale de y' = 2y est y(x) = Ce^{2x}, où C est une constante réelle.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La solution générale de y' = 2y est bien y(x) = Ce^{2x}, car en dérivant, on obtient 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":35592,"question":"Quelle méthode permet de résoudre une équation différentielle de la forme y' = f(x)g(y) ?","option_a":"Méthode de variation de la constante","option_b":"Méthode de séparation des variables","option_c":"Méthode des coefficients indéterminés","option_d":"Méthode d'Euler","option_e":"","option_f":"","bonne_reponse":"b","explication":"La méthode de séparation des variables consiste à écrire l'équation sous la forme dy\/g(y) = f(x)dx et à intégrer les deux membres.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Méthode de variation de la constante\", \"b\": \"Méthode de sépara","_debug_options_count":4},{"id":35593,"question":"Soit l'équation différentielle y'' - 3y' + 2y = 0. Quelle est sa solution générale ?","option_a":"y(x) = C1e^x + C2e^{2x}","option_b":"y(x) = C1e^x + C2e^{-x}","option_c":"y(x) = C1e^{2x} + C2e^{x}","option_d":"y(x) = C1e^{-x} + C2e^{-2x}","option_e":"","option_f":"","bonne_reponse":"c","explication":"L'équation caractéristique associée est r² - 3r + 2 = 0, dont les racines sont r1=1 et r2=2. La solution générale est donc y(x) = C1e^x + C2e^{2x}.","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(x) = C1e^x + C2e^{2x}\", \"b\": \"y(x) = C1e^x + C2e^{-x}\", \"c\": \"y","_debug_options_count":4},{"id":35594,"question":"L'équation différentielle y' = y² est une équation différentielle linéaire.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'équation y' = y² est non linéaire car le terme y² fait intervenir un produit de y par lui-même, ce qui viole la linéarité.","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":35595,"question":"Quelle est la solution générale de l'équation différentielle y' + y = e^{-x} ?","option_a":"y(x) = Ce^{-x} + xe^{-x}","option_b":"y(x) = Ce^{-x}","option_c":"y(x) = Ce^{-x} + x","option_d":"y(x) = Ce^{-x} + e^{-x}","option_e":"","option_f":"","bonne_reponse":"a","explication":"La solution générale est y(x) = Ce^{-x} + xe^{-x}, où C est une constante. La solution particulière xe^{-x} est obtenue par la méthode de 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\": \"a\", \"options\": {\"a\": \"y(x) = Ce^{-x} + xe^{-x}\", \"b\": \"y(x) = Ce^{-x}\", \"c\": \"y(x) = Ce","_debug_options_count":4},{"id":35596,"question":"Soit l'équation différentielle y'' + 4y = 0. Quelle est la nature de ses solutions ?","option_a":"Exponentielles réelles","option_b":"Trigonométriques","option_c":"Logarithmiques","option_d":"Polynomiales","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'équation caractéristique r² + 4 = 0 a pour racines r = ±2i, donc les solutions sont de la forme y(x) = C1cos(2x) + C2sin(2x), qui sont des fonctions trigonométriques.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Exponentielles réelles\", \"b\": \"Trigonométriques\", \"c\": \"Logarit","_debug_options_count":4},{"id":35597,"question":"L'équation différentielle y' = 3y + 2 admet une solution constante.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une solution constante y = k vérifie 0 = 3k + 2, donc k = -2\/3. Ainsi, y(x) = -2\/3 est une solution constante de 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":35598,"question":"Quelle est la solution de l'équation différentielle y' = 5 avec la condition initiale y(0) = 3 ?","option_a":"y(x) = 5x + 3","option_b":"y(x) = 5x","option_c":"y(x) = 3","option_d":"y(x) = 5x + C","option_e":"","option_f":"","bonne_reponse":"a","explication":"La solution générale est y(x) = 5x + C. Avec y(0) = 3, on obtient C = 3, donc y(x) = 5x + 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\": \"y(x) = 5x + 3\", \"b\": \"y(x) = 5x\", \"c\": \"y(x) = 3\", \"d\": \"y(x) = 5","_debug_options_count":4},{"id":35599,"question":"L'équation différentielle y'' - 2y' + y = 0 admet une solution double.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'équation caractéristique r² - 2r + 1 = 0 a une racine double r = 1, donc la solution générale est y(x) = (C1 + C2x)e^x, qui est une solution double.","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}]