Quiz — 4. equations diffA rentielle -2020.docx

Question 1 sur 10 20:00

[{"id":31680,"question":"Quelle est la forme générale d'une équation différentielle linéaire du second ordre à coefficients constants ?","option_a":"A. y'' + a(x)y' + b(x)y = f(x)","option_b":"B. y'' + ay' + by = 0","option_c":"C. y'' + ay' + by = f(x)","option_d":"D. y'' + a(x)y' = b(x)y","option_e":"","option_f":"","bonne_reponse":"c","explication":"La forme générale est y'' + ay' + by = f(x), où a et b sont des constantes et f(x) est le second membre (nul pour une équation homogène).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"A. y'' + a(x)y' + b(x)y = f(x)\", \"b\": \"B. y'' + ay' + by = 0\", \"c","_debug_options_count":4},{"id":31681,"question":"La solution générale d'une équation différentielle homogène du second ordre est une combinaison linéaire de deux solutions indépendantes.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai. La solution générale s'écrit y(x) = C1y1(x) + C2y2(x), où y1 et y2 sont deux solutions linéairement indépendantes 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":31682,"question":"Pour une équation différentielle y'' - 4y = 0, quelle est la forme des solutions ?","option_a":"A. y(x) = C1e^(2x) + C2e^(-2x)","option_b":"B. y(x) = C1cos(2x) + C2sin(2x)","option_c":"C. y(x) = C1e^(x) + C2e^(-x)","option_d":"D. y(x) = C1x + C2","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'équation caractéristique associée est r² - 4 = 0, dont les racines sont r = 2 et r = -2. Les solutions sont donc y(x) = C1e^(2x) + C2e^(-2x).","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(x) = C1e^(2x) + C2e^(-2x)\", \"b\": \"B. y(x) = C1cos(2x) + C2si","_debug_options_count":4},{"id":31683,"question":"Si les racines de l'équation caractéristique sont complexes (α ± iβ), quelle est la forme des solutions ?","option_a":"A. y(x) = C1e^(αx) + C2e^(-αx)","option_b":"B. y(x) = e^(αx)(C1cos(βx) + C2sin(βx))","option_c":"C. y(x) = C1cos(βx) + C2sin(βx)","option_d":"D. y(x) = C1x + C2","option_e":"","option_f":"","bonne_reponse":"b","explication":"Les solutions s'expriment sous la forme y(x) = e^(αx)(C1cos(βx) + C2sin(βx)) pour des racines complexes α ± iβ.","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. y(x) = C1e^(αx) + C2e^(-αx)\", \"b\": \"B. y(x) = e^(αx)(C1cos(","_debug_options_count":4},{"id":31684,"question":"La méthode de variation des constantes s'applique uniquement aux équations différentielles du premier ordre.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Faux. La méthode de variation des constantes peut s'étendre aux équations différentielles d'ordre supérieur, y compris le second 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},{"id":31685,"question":"Quelle est la solution générale de l'équation différentielle y'' + y = 0 ?","option_a":"A. y(x) = C1e^(x) + C2e^(-x)","option_b":"B. y(x) = C1cos(x) + C2sin(x)","option_c":"C. y(x) = C1x + C2","option_d":"D. y(x) = C1e^(ix) + C2e^(-ix)","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'équation caractéristique est r² + 1 = 0, dont les racines sont r = ±i. Les solutions sont donc y(x) = C1cos(x) + C2sin(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. y(x) = C1e^(x) + C2e^(-x)\", \"b\": \"B. y(x) = C1cos(x) + C2sin(x","_debug_options_count":4},{"id":31686,"question":"Pour résoudre une équation différentielle non homogène, quelle méthode est la plus adaptée si le second membre est une constante ?","option_a":"A. Méthode de variation des constantes","option_b":"B. Méthode des coefficients indéterminés","option_c":"C. Intégration directe","option_d":"D. Changement de variable","option_e":"","option_f":"","bonne_reponse":"b","explication":"La méthode des coefficients indéterminés est la plus adaptée pour un second membre constant ou polynomial.","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. Méthode de variation des constantes\", \"b\": \"B. Méthode des c","_debug_options_count":4},{"id":31687,"question":"Si une équation différentielle a pour solution générale y(x) = C1e^(2x) + C2e^(-x), quelle est son équation caractéristique associée ?","option_a":"A. r² + r - 2 = 0","option_b":"B. r² - r - 2 = 0","option_c":"C. r² + 2r - 1 = 0","option_d":"D. r² - 2r + 1 = 0","option_e":"","option_f":"","bonne_reponse":"a","explication":"Les solutions e^(2x) et e^(-x) correspondent aux racines r = 2 et r = -1. L'équation caractéristique est donc (r - 2)(r + 1) = r² - 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\": \"A. r² + r - 2 = 0\", \"b\": \"B. r² - r - 2 = 0\", \"c\": \"C. r² + 2r","_debug_options_count":4},{"id":31688,"question":"La condition initiale y(0) = 3 et y'(0) = -1 permet de déterminer les constantes C1 et C2 dans la solution générale.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai. Les conditions initiales fournissent un système de deux équations pour résoudre C1 et C2.","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":31689,"question":"Quelle est la solution particulière de l'équation y'' - 3y' + 2y = 2x + 1 ?","option_a":"A. y_p(x) = Ax + B","option_b":"B. y_p(x) = Ax² + Bx + C","option_c":"C. y_p(x) = Axe^(x)","option_d":"D. y_p(x) = A","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le second membre est un polynôme de degré 1. On cherche une solution particulière sous la forme y_p(x) = Ax + B.","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_p(x) = Ax + B\", \"b\": \"B. y_p(x) = Ax² + Bx + C\", \"c\": \"C. y","_debug_options_count":4}]