First step if the equation is x\frac{dy}{dx}+3y=x^2?

Divide by x to get \frac{dy}{dx}+\frac3xy=x.

What is the difference between a general and a particular solution?

A general solution has an arbitrary constant; a particular solution fixes it using a given condition.

What do solution curves of a first-order equation do?

Form a family that never crosses, one curve through each point.

Terminal speed from \frac{dv}{dt}=10-0.4v?

Set \frac{dv}{dt}=0: v=25 m s^{-1}.

When is an integrating factor appropriate?

When the equation is linear in y, in the form \frac{dy}{dx}+Py=Q, and not simply separable.

First-order differential equations and integrating factorsEdexcel A-Level Further Maths: Flashcards

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Standard form of a linear first-order differential equation?

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Standard form of a linear first-order differential equation?: $\frac{dy}{dx}+P(x)y=Q(x)$
Formula for the integrating factor?: $I=e^{\int P(x)\,dx}$
What does multiplying by $I$ do to the left-hand side?: It becomes $\frac{d}{dx}(Iy)$ .
After multiplying by $I$ , what do you integrate?: $Iy=\int IQ\,dx$
First step if the equation is $x\frac{dy}{dx}+3y=x^2$ ?: Divide by $x$ to get $\frac{dy}{dx}+\frac3xy=x$ .
Integrating factor for $\frac{dy}{dx}+\frac3xy=x$ ( $x>0$ )?: $e^{3\ln x}=x^3$
Integrating factor for $\frac{dy}{dx}+y\tan x=\cos x$ ?: $e^{\ln\sec x}=\sec x$
General solution of $\frac{dy}{dx}+2y=6$ ?: $y=3+ce^{-2x}$
What is the difference between a general and a particular solution?: A general solution has an arbitrary constant; a particular solution fixes it using a given condition.
What do solution curves of a first-order equation do?: Form a family that never crosses, one curve through each point.
Basic mixing model for the mass $m$ of salt in a tank?: $\frac{dm}{dt}=\text{rate in}-\text{rate out}$
Terminal speed from $\frac{dv}{dt}=10-0.4v$ ?: Set $\frac{dv}{dt}=0$ : $v=25$ m s $^{-1}$ .
When is an integrating factor appropriate?: When the equation is linear in $y$ , in the form $\frac{dy}{dx}+Py=Q$ , and not simply separable.

Exam questions on First-order differential equations and integrating factors

Consider the differential equation $\frac{dy}{dx}+\frac{2}{x}y=x^2$ for $x>0$ .
Given that $y=1$ when $x=1$ , find $y$ in terms of $x$ .2 marks
A curve satisfies $\frac{dy}{dx}+y\tan x=\cos x$ for $-\frac{\pi}{2}<x<\frac{\pi}{2}$ .
The curve passes through the point $\left(\frac{\pi}{3},\frac{\pi}{3}\right)$ . Find $y$ in terms of $x$ .2 marks
A tank holds 200 litres of well-stirred brine containing 10 kg of salt at time $t=0$ , where $t$ is in minutes. Brine of concentration 0.5 kg per litre flows in at 4 litres per minute, and the well-stirred mixture flows out at 4 litres per minute. Let $m$ kg be the mass of salt in the tank at time $t$ .
Show that $\frac{dm}{dt}+\frac{m}{50}=2$ .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).