Assertion (A): The solution of dy/dx + y = xe^-x with y = 1 at x = 0 is y = (x^2/2 + 1)e^-x. Reason (R): The integrating factor of dy/dx + y = xe^-x is e^-x.

Question 108

MediumOriginal question

Assertion (A): The solution of $\frac{dy}{dx} + y = xe^{-x}$ with $y = 1$ at $x = 0$ is $y = \left(\frac{x^{2}}{2} + 1\right)e^{-x}$ .
Reason (R): The integrating factor of $\frac{dy}{dx} + y = xe^{-x}$ is $e^{-x}$ .

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 C

Correct answer: C

$P = 1$ , so the factor is $e^{x}$ and R is false: $ye^{x} = \int x\,dx = \frac{x^{2}}{2} + C$ .
$y(0) = C = 1$ , so $y = \left(\frac{x^{2}}{2} + 1\right)e^{-x}$ and A is true.

Worth noting

The factor comes from $P = 1$ , not from $Q = xe^{-x}$ , so it is $e^{x}$ .

Concept: A particular solution of a linear equation
Typical timeTypical: 90 s
Your timeYours: —

The rule this testsSolving linear equations

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