Quiz — cours_equadiff.pdf

Question 1 sur 10 20:00

[{"id":87353,"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":87354,"question":"Une équation différentielle est dite homogène si son second membre est nul.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une équation différentielle est homogène lorsque son second membre est égal à zéro.","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":87355,"question":"Quelle méthode utilise-t-on pour résoudre une équation différentielle linéaire du second ordre à coefficients constants ?","option_a":"La méthode de variation de la constante","option_b":"La méthode de séparation des variables","option_c":"La méthode de résolution de l'équation caractéristique","option_d":"La méthode de substitution","option_e":"","option_f":"","bonne_reponse":"c","explication":"Pour résoudre une équation différentielle linéaire du second ordre à coefficients constants, on utilise la méthode de résolution 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\": \"c\", \"options\": {\"a\": \"La méthode de variation de la constante\", \"b\": \"La méthode de s","_debug_options_count":4},{"id":87356,"question":"Quelle est la solution générale de l'équation différentielle y' = 2y ?","option_a":"y = Ce^(2x)","option_b":"y = Cx^2","option_c":"y = 2x + C","option_d":"y = C*e^(-2x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La solution générale de l'équation différentielle y' = 2y est y = Ce^(2x), où C est une constante réelle.","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 = 2x + C\", \"d\": \"y = C*e^(","_debug_options_count":4},{"id":87357,"question":"L'équation différentielle y'' - 3y' + 2y = 0 est une équation homogène.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'équation y'' - 3y' + 2y = 0 est homogène car son second membre est 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":87358,"question":"Quelle est la solution particulière de l'équation différentielle y' + y = x^2 ?","option_a":"y = x^2 - 2x + 2","option_b":"y = -x^2 + 2x - 2","option_c":"y = x^2 + 2x + 2","option_d":"y = -x^2 - 2x + 2","option_e":"","option_f":"","bonne_reponse":"b","explication":"La solution particulière de l'équation y' + y = x^2 est y = -x^2 + 2x - 2, 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\": \"b\", \"options\": {\"a\": \"y = x^2 - 2x + 2\", \"b\": \"y = -x^2 + 2x - 2\", \"c\": \"y = x^2 + 2x +","_debug_options_count":4},{"id":87359,"question":"Quelle est la solution générale de l'équation différentielle y'' + 4y = 0 ?","option_a":"y = C1*cos(2x) + C2*sin(2x)","option_b":"y = C1*e^(2x) + C2*e^(-2x)","option_c":"y = C1*x + C2","option_d":"y = C1*e^(4x) + C2*e^(-4x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La solution générale de l'équation différentielle y'' + 4y = 0 est y = C1*cos(2x) + C2*sin(2x), où C1 et C2 sont des constantes réelles.","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 = C1*cos(2x) + C2*sin(2x)\", \"b\": \"y = C1*e^(2x) + C2*e^(-2x)\", ","_debug_options_count":4},{"id":87360,"question":"La méthode de variation de la constante s'applique uniquement aux équations différentielles linéaires du premier ordre.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La méthode de variation de la constante s'applique aussi aux équations différentielles linéaires d'ordre supérieur, notamment du 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":87361,"question":"Quelle est la solution générale de l'équation différentielle y' - 5y = 0 ?","option_a":"y = Ce^(5x)","option_b":"y = Cx^5","option_c":"y = 5x + C","option_d":"y = C*e^(-5x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La solution générale de l'équation différentielle y' - 5y = 0 est y = Ce^(5x), où C est une constante réelle.","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^(5x)\", \"b\": \"y = Cx^5\", \"c\": \"y = 5x + C\", \"d\": \"y = C*e^(","_debug_options_count":4},{"id":87362,"question":"Quelle est la solution particulière de l'équation différentielle y'' - y = e^x ?","option_a":"y = (x\/2)*e^x","option_b":"y = -x*e^x","option_c":"y = (x^2\/2)*e^x","option_d":"y = x*e^(-x)","option_e":"","option_f":"","bonne_reponse":"b","explication":"La solution particulière de l'équation y'' - y = e^x est y = -x*e^x, obtenue par la méthode de variation des paramètres.","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 = (x\/2)*e^x\", \"b\": \"y = -x*e^x\", \"c\": \"y = (x^2\/2)*e^x\", \"d\": \"","_debug_options_count":4}]