What does multiplying by I achieve?

The left side becomes \frac{d}{dx}(Iy), which can be integrated.

Do you include a constant when finding I?

No; any one antiderivative of P will do.

Integrating factor for \frac{dy}{dx}+y\tan x=\cos x?

e^{\int\tan x\,dx}=e^{\ln\sec x}=\sec x

What must you do first to x\frac{dy}{dx}+3y=x^3?

Divide by x to get coefficient 1 on \frac{dy}{dx}.

What is the difference between general and particular solutions?

The general solution has constant c; the particular solution uses a condition to find c.

Which equations are unsuitable for the method?

Non-linear ones, such as \frac{dy}{dx}+y^2=x.

How do you find the maximum of a solution v(t)?

Solve \frac{dv}{dt}=0, then substitute into v.

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

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What is the standard form of a first-order linear equation?

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What is the standard form of a first-order linear equation?: $\frac{dy}{dx}+P(x)y=Q(x)$
State the integrating factor.: $I=e^{\int P(x)\,dx}$
What does multiplying by $I$ achieve?: The left side becomes $\frac{d}{dx}(Iy)$ , which can be integrated.
What does the equation become after multiplying by $I$ ?: $\frac{d}{dx}(Iy)=IQ$
Do you include a constant when finding $I$ ?: No; any one antiderivative of $P$ will do.
Simplify $e^{3\ln x}$ .: $x^3$
Simplify $e^{-\ln x}$ .: $\frac1x$
Integrating factor for $\frac{dy}{dx}+y\tan x=\cos x$ ?: $e^{\int\tan x\,dx}=e^{\ln\sec x}=\sec x$
Integrating factor for $\frac{dy}{dx}-2y=e^{3x}$ ?: $e^{-2x}$
What must you do first to $x\frac{dy}{dx}+3y=x^3$ ?: Divide by $x$ to get coefficient 1 on $\frac{dy}{dx}$ .
What is the difference between general and particular solutions?: The general solution has constant $c$ ; the particular solution uses a condition to find $c$ .
Which equations are unsuitable for the method?: Non-linear ones, such as $\frac{dy}{dx}+y^2=x$ .
What is $\frac{dv}{dt}$ in kinematics?: Acceleration.
How do you find the maximum of a solution $v(t)$ ?: Solve $\frac{dv}{dt}=0$ , then substitute into $v$ .

Exam questions on First order equations and integrating factors

$\frac{dy}{dx}+\frac{3}{x}y=x^2$ , for $x>0$ .
Find the general solution, giving $y$ in terms of $x$ .2 marks
$\frac{dy}{dx}-2y=e^{3x}$ .
Given that $y=3$ when $x=0$ , find $y$ in terms of $x$ .2 marks
$\frac{dy}{dx}+y\tan x=\cos x$ , for $-\frac{\pi}{2}<x<\frac{\pi}{2}$ .
Show that the general solution is $y=(x+c)\cos x$ .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).