6-4b6a-8cb1-abbcd3d5379c","title":"1.9.31 Formulate first order differential equations (A-level only) Flashcards","topicId":"f95a6e84-8009-4e39-a3a3-b01a97867740","topicSlug":"formulate-first-order-differential-equations","cards":[{"id":"1638143c-e5d5-42a7-a9bf-299268920518","cardType":"cloze","front":"The rate of change of $y$ with respect to $x$ is written $\\text{[...]}$.","back":"The rate of change of $y$ with respect to $x$ is written $\\boxed{\\frac{dy}{dx}}$.","points":null,"mcqOptions":null},{"id":"77a17fcf-aa2e-4b6a-9158-5ba75965a49c","cardType":"basic","front":"What does $\\frac{dP}{dt}$ represent for a population $P$?","back":"The instantaneous rate of population change with time.","points":1,"mcqOptions":null},{"id":"4d29f2f6-3d86-4cca-bcbb-827a52fd7028","cardType":"cloze","front":"The units of a derivative are [ ].","back":"The units of a derivative are dependent-variable units divided by independent-variable units.","points":null,"mcqOptions":null},{"id":"26993ad3-d3d0-4a89-be74-14ee9edab8cc","cardType":"basic","front":"Why is $\\frac{dy}{dx}=3x-y$ first order?","back":"Its highest derivative is $\\frac{dy}{dx}$.","points":2,"mcqOptions":null},{"id":"082fa00e-63ad-4794-8167-d77bbb0ea917","cardType":"basic","front":"Formulate the equation when $M$ decreases by 8 units per hour.","back":"$\\frac{dM}{dt}=-8$","points":1,"mcqOptions":null},{"id":"a00f84cd-3ea7-4a8c-9a19-5ddc71fa8a20","cardType":"basic","front":"Formulate the equation when the growth rate of $y$ is proportional to $y$.","back":"$\\frac{dy}{dx}=ky$, where $k>0$.","points":1,"mcqOptions":null},{"id":"ccfe9981-eccf-4c97-82c5-daf14fb40cbf","cardType":"basic","front":"A quantity $Q$ decreases at a rate proportional to $Q^2$. Formulate its equation using $k>0$.","back":"$\\frac{dQ}{dt}=-kQ^2$","points":1,"mcqOptions":null},{"id":"1209f25f-ff91-45c5-b22f-a6ad7b1f008a","cardType":"cloze","front":"For competing processes, $\\text{[...]}$.","back":"For competing processes, $\\boxed{\\text{net rate}=\\text{rates gained}-\\text{rates lost}}$.","points":null,"mcqOptions":null},{"id":"a7a422df-ba24-4cbc-84f9-780f9d0a4348","cardType":"basic","front":"A population grows at rate $0.04P$ while 120 individuals are removed yearly. Formulate the model.","back":"$\\frac{dP}{dt}=0.04P-120$","points":2,"mcqOptions":null},{"id":"1bc63e5c-c88c-4634-8bf3-75569d29ca7a","cardType":"basic","front":"Births occur at rate $aN$ and deaths at rate $bN^2$. Formulate the population model.","back":"$\\frac{dN}{dt}=aN-bN^2$","points":2,"mcqOptions":null},{"id":"018e7f6a-dfc2-445a-92ef-3a8868072196","cardType":"basic","front":"Why is $\\frac{dN}{dt}=aN-bN^2$ first order despite containing $N^2$?","back":"Order depends on the highest derivative, not the power of $N$.","points":2,"mcqOptions":null},{"id":"568c9dbc-4fbe-4f94-84b8-83c29f039e6b","cardType":"basic","front":"Formulate a cooling model for temperature $T$ in a room at $20^\\circ\\text{C}$, using $k>0$.","back":"$\\frac{dT}{dt}=-k(T-20)$","points":2,"mcqOptions":null},{"id":"3bb09a9c-5eb2-47f0-bb4b-c4774e9e353a","cardType":"basic","front":"What sign does $\\frac{dy}{dt}$ have in $\\frac{dy}{dt}=k(A-y)$ when $yPositive, so $y$ moves towards $A$.","points":1,"mcqOptions":null},{"id":"12369458-9ace-4af5-95a6-642871eca493","cardType":"cloze","front":"A population initially equal to 500 has initial condition $\\text{[...]}$.","back":"A population initially equal to 500 has initial condition $\\boxed{P(0)=500}$.","points":null,"mcqOptions":null},{"id":"f400eec8-6347-4beb-849f-78d5f2b60c47","cardType":"cloze","front":"A dissolved substance enters or leaves at rate $\\text{[...]}$.","back":"A dissolved substance enters or leaves at rate $\\boxed{\\text{volume flow rate}\\times\\text{concentration}}$.","points":null,"mcqOptions":null},{"id":"321d1b