What is the integrating factor of xdy/dx + (1 + x)y = x^2, for x > 0?

Homogeneous & Linear Differential Equations · Question 88 of 120

Question 88

HardOriginal question

What is the integrating factor of $x\frac{dy}{dx} + (1 + x)y = x^{2}$ , for $x > 0$ ?

x + e^{x}

Correct answerYour answer

e^{x}

Correct answerYour answer

e^{x + 1}

Correct answerYour answer

x e^{x}

Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer D

Correct answer: D

Divide by $x$ : $P = \frac{1 + x}{x} = \frac{1}{x} + 1$ , so $\int P\,dx = \log x + x$ .
The factor is $e^{\log x + x} = e^{\log x} \cdot e^{x} = x\,e^{x}$ .

Worth noting

$x + e^{x}$ splits $e^{\log x + x}$ into a sum; $e$ raised to a sum is a product.

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

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