5.14 Differential equations: separation of variablesIB Maths: Applications and Interpretation HL: Flashcards

What these 13 flashcards ask

What does "proportional to" mean in a differential equation?
How do you model decay at a rate proportional to m?
What is separation of variables?
General solution of \frac{dy}{dx}=ky?
How is the constant A obtained from \ln y=kx+c?
What is a general solution?
What is a particular solution?
Solve \frac{dy}{dx}=2y with y=5 when x=0.
What is the solution of Newton's law of cooling, \frac{dT}{dt}=-k(T-T0)?
Doubling time for \frac{dG}{dt}=kG?
How do you find k from a later value such as G=80 at t=4?
Where does the constant of integration go when separating variables?
What should you do after finding a model?

Exam questions on 5.14 Differential equations: separation of variables

The mass $G$ grams of algae in a pond at time $t$ days grows at a rate proportional to $G$ , so $\frac{dG}{dt}=kG$ where $k>0$ is a constant. Initially $G=50$ , and when $t=4$ , $G=80$ .
Find the mass of algae after $10$ days, using the unrounded value of $k$ .2 marks
A sample of a radioactive substance has mass $m$ mg at time $t$ years. The mass decreases at a rate proportional to $m$ . Initially $m=200$ , and after $5$ years $m=150$ .
Find the value of $k$ .2 marks
A boat's engine is switched off at $t=0$ . For $t\ge0$ the speed $v$ m s $^{-1}$ of the boat satisfies $\frac{dv}{dt}=-0.05v^2$ , with $v=10$ when $t=0$ .
Show that $v=\frac{20}{t+2}$ .3 marks

Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).