Separable differential equationsAQA A-Level Maths: Flashcards

What these 12 flashcards ask

What makes \frac{dy}{dx}=f(x)g(y) separable?
State the method of separation of variables.
Solve \frac{dy}{dx}=2xy, y0 (general solution).
What is the difference between a general and a particular solution?
How do you deal with \ln y=x^2+c when making y the subject?
Factorise to separate: \frac{dy}{dx}=6x+3xy.
Solve \frac{dv}{dt}=-kv with v=v0 when t=0.
Newton's law of cooling in differential form?
Which derivatives link kinematics to differential equations?
What does \theta=20+70e^{-kt} predict as t\to\infty?
Name two common limitations of a differential equation model.
What is \int\frac1y\,dy, and when can you drop the modulus?

Exam questions on Separable differential equations

A curve satisfies the differential equation $\frac{dy}{dx}=2xy$ , where $y>0$ .
Given that $y=3$ when $x=0$ , find $y$ in terms of $x$ .2 marks
A particle moves in a straight line. Its velocity $v$ m s $^{-1}$ at time $t$ seconds satisfies $\frac{dv}{dt}=-0.5v$ , and $v=8$ when $t=0$ .
Explain, using your solution, why the model predicts that the particle never comes to rest, and suggest why this is unrealistic.2 marks
A curve satisfies the differential equation $\frac{dy}{dx}=6x+3xy$ for $x\ge0$ .
Show that the general solution can be written $y=Ae^{\frac32x^2}-2$ , where $A$ is a constant.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).