Assertion (A): The integrating factor of dy/dx + Py = Q is e^∫ P dx. Reason (R): Multiplying by e^∫ P dx turns the left side into d/dx(y e^∫ P dx).

Question 68

MediumOriginal question

Assertion (A): The integrating factor of $\frac{dy}{dx} + Py = Q$ is $e^{\int P\,dx}$ .
Reason (R): Multiplying by $e^{\int P\,dx}$ turns the left side into $\frac{d}{dx}\left(y\,e^{\int P\,dx}\right)$ .

Both A and R are true, and R is the correct explanation of A.Correct answerYour answerBoth A and R are true, but R is not the correct explanation of A.Correct answerYour answerA is true, but R is false.Correct answerYour answerA is false, but R is true.Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer A

Correct answer: A

A is true, and R is the reason: the product rule gives $e^{\int P\,dx}(y' + Py)$ .
Then integrating both sides solves the equation.

Concept: Why the integrating factor works
Typical timeTypical: 90 s
Your timeYours: —

The rule this testsLinear equations and the integrating factor

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Ten questions from Homogeneous & Linear Differential Equations, at your own pace.

Assertion (A): The integrating factor of dydx+Py=Q\frac{dy}{dx} + Py = Qdxdy​+Py=Q is e∫P dxe^{\int P\,dx}e∫Pdx.Reason (R): Multiplying by e∫P dxe^{\int P\,dx}e∫Pdx turns the left side into ddx(y e∫P dx)\frac{d}{dx}\left(y\,e^{\int P\,dx}\right)dxd​(ye∫Pdx).

Explanation

Assertion (A): The integrating factor of $\frac{dy}{dx} + Py = Q$ is $e^{\int P\,dx}$ .
Reason (R): Multiplying by $e^{\int P\,dx}$ turns the left side into $\frac{d}{dx}\left(y\,e^{\int P\,dx}\right)$ .