For $x > 0$ , which of these equations has the integrating factor $x^{2}$ ?

x\frac{dy}{dx} - 2y = e^{x}

Correct answerYour answer

x\frac{dy}{dx} + 2y = e^{x}

Correct answerYour answer

\frac{dy}{dx} + 2xy = e^{x}

Correct answerYour answer

x^{2}\frac{dy}{dx} + 2y = e^{x}

Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer B

Correct answer: B

$x\frac{dy}{dx} + 2y = e^{x}$ , divided by $x$ , has $P = \frac{2}{x}$ , so its factor is $e^{2\log x} = x^{2}$ .
The others have $P = -\frac{2}{x}$ , $2x$ and $\frac{2}{x^{2}}$ , with factors $\frac{1}{x^{2}}$ , $e^{x^{2}}$ and $e^{-2/x}$ .

Worth noting

$x\frac{dy}{dx} - 2y = e^{x}$ has $P = -\frac{2}{x}$ , whose factor is $\frac{1}{x^{2}}$ , not $x^{2}$ .

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

For x>0x > 0x>0, which of these equations has the integrating factor x2x^{2}x2?