Which of the following statements are true?
I. An integrating factor $e^{\int P\,dx}$ is never 0.
II. If $P$ is a constant $a$ , the integrating factor is $e^{ax}$ .
III. The integrating factor depends on $Q$ as well as on $P$ .
IV. For $x > 0$ , $\frac{dy}{dx} + \frac{y}{x} = Q$ and $x\frac{dy}{dx} + y = xQ$ have the same integrating factor.

All four: I, II, III and IVCorrect answerYour answerI, II and IV onlyCorrect answerYour answerI and II onlyCorrect answerYour answerII and IV onlyCorrect answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer B

Correct answer: B

I: a power of $e$ is always positive. II: $e^{\int a\,dx} = e^{ax}$ . IV: dividing the second equation by $x$ gives the first, so both have $P = \frac{1}{x}$ and factor $x$ .
III is false: the factor is $e^{\int P\,dx}$ , built from $P$ alone.

Worth noting

$Q$ enters only after the factor is found, in $\int Q \cdot e^{\int P\,dx}\,dx$ .

Concept: Facts about the integrating factor
Typical timeTypical: 120 s
Your timeYours: —

The rule this testsLinear equations and the integrating factor

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