First step for x\frac{dy}{dx}+3y=x^2?

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

Why is no constant needed when finding I?

Any constant only multiplies both sides of the equation and cancels.

Where does the constant C go in the final answer?

Divide through by I so C is divided too, e.g. Ce^{-3x}.

Tank mixing: \frac{dm}{dt} equals?

Rate of salt in minus rate of salt out.

First order linear differential equationsEdexcel International A Level Further Maths: Flashcards

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Standard form of a first order linear equation?

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Standard form of a first order linear equation?: $\frac{dy}{dx}+Py=Q$ , with $P,Q$ functions of $x$ .
Integrating factor for $\frac{dy}{dx}+Py=Q$ ?: $I=e^{\int P\,dx}$ .
What is $\frac{d}{dx}(yI)$ after multiplying by $I$ ?: $QI$ .
Integrating factor for $\frac{dy}{dx}+3y=\ldots$ ?: $e^{3x}$ .
First step for $x\frac{dy}{dx}+3y=x^2$ ?: Divide by $x$ : $\frac{dy}{dx}+\frac3xy=x$ .
Simplify $e^{k\ln x}$ .: $x^k$ .
Integrating factor when $P=\tan x$ ?: $\sec x$ .
Integrating factor when $P=\cot x$ ?: $\sin x$ .
Integrating factor when $P=\frac2x$ ( $x>0$ )?: $x^2$ .
Integrating factor when $P=-\frac1x$ ( $x>0$ )?: $\frac1x$ .
Why is no constant needed when finding $I$ ?: Any constant only multiplies both sides of the equation and cancels.
Where does the constant $C$ go in the final answer?: Divide through by $I$ so $C$ is divided too, e.g. $Ce^{-3x}$ .
Tank mixing: $\frac{dm}{dt}$ equals?: Rate of salt in minus rate of salt out.
Rate of salt out for volume $V$ and outflow $r$ ?: $r\times\frac{m}{V}$ .

Exam questions on First order linear differential equations

Consider the differential equation $\frac{dy}{dx}+3y=e^{2x}$ .
Find the general solution.2 marks
Consider the differential equation $x\frac{dy}{dx}-2y=x^3$ for $x>0$ .
Find the general solution.2 marks
Consider the differential equation $\frac{dy}{dx}+y\tan x=\sec x$ for $0<x<\frac{\pi}{2}$ .
Show that the integrating factor is $\sec x$ .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).