How is a constant A introduced when solving \ln y=x^2+c?

y=e^{x^2+c}=Ae^{x^2}, with A=e^c.

What does a particular solution require?

A condition such as y=y0 at x=x0 to find the constant.

'Rate of change of \theta is proportional to \theta-20'?

\frac{d\theta}{dt}=k(\theta-20), with k<0 for cooling.

Solution curves of \frac{dy}{dx}=-\frac xy?

Circles x^2+y^2=k centred at the origin.

Why do members of a solution family never cross?

Through each point there is exactly one solution of the differential equation.

How do you find stationary points of a solution curve?

Set \frac{dy}{dx}=0 in the differential equation.

Separable differential equations and families of solutionsEdexcel International A Level Further Maths: Flashcards

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How do you solve $\frac{dy}{dx}=f(x)g(y)$?

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How do you solve $\frac{dy}{dx}=f(x)g(y)$ ?: Separate: $\int\frac{1}{g(y)}\,dy=\int f(x)\,dx$ .
General solution of $\frac{dy}{dx}=ky$ ?: $y=Ae^{kx}$ .
General solution of $\frac{dy}{dx}=2xy$ , $y>0$ ?: $y=Ae^{x^2}$ .
How is a constant $A$ introduced when solving $\ln y=x^2+c$ ?: $y=e^{x^2+c}=Ae^{x^2}$ , with $A=e^c$ .
What does a particular solution require?: A condition such as $y=y_0$ at $x=x_0$ to find the constant.
'Rate of decrease of $m$ is proportional to $m$ ' gives?: $\frac{dm}{dt}=-km$ with $k>0$ .
'Rate of change of $\theta$ is proportional to $\theta-20$ '?: $\frac{d\theta}{dt}=k(\theta-20)$ , with $k<0$ for cooling.
Newton cooling solution with room temperature $T_0$ ?: $\theta=T_0+Ae^{-kt}$ .
Integral of $\frac{1}{y^2}$ with respect to $y$ ?: $-\frac1y$ .
Solution of $x\frac{dy}{dx}=y^2$ through $(1,1)$ ?: $y=\frac{1}{1-\ln x}$ .
Solution curves of $\frac{dy}{dx}=-\frac xy$ ?: Circles $x^2+y^2=k$ centred at the origin.
Why do members of a solution family never cross?: Through each point there is exactly one solution of the differential equation.
How do you find stationary points of a solution curve?: Set $\frac{dy}{dx}=0$ in the differential equation.

Exam questions on Separable differential equations and families of solutions

A curve $C$ passes through the point $(0,3)$ and satisfies the differential equation $\frac{dy}{dx}=2xy$ , where $y>0$ .
Show that $C$ has a minimum point at $(0,3)$ .2 marks
A drink cools in a room at a constant temperature of $20\,^\circ\text{C}$ . At time $t$ minutes its temperature is $\theta\,^\circ\text{C}$ , and the rate of decrease of $\theta$ is proportional to $\theta-20$ , with constant of proportionality $k>0$ . Initially $\theta=80$ .
After $5$ minutes the temperature is $50\,^\circ\text{C}$ . Find the exact value of $k$ .2 marks
A curve satisfies the differential equation $\frac{dy}{dx}=xe^{-y}$ .
Find the general solution, giving $e^y$ in terms of $x$ .3 marks

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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).