What is the general solution of xdy/dx + 2y = x^2?

Homogeneous & Linear Differential Equations · Question 112 of 120

Question 112

HardOriginal question

What is the general solution of $x\frac{dy}{dx} + 2y = x^{2}$ ?

y = \frac{x^{2}}{4} + \frac{C}{x^{2}}

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y = \frac{x^{2}}{2} + \frac{C}{x^{2}}

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y = \frac{x^{2}}{4} + Cx^{2}

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y = \frac{x^{4}}{4} + C

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CorrectNot quiteAnswer A

Correct answer: A

Divide by $x$ : $\frac{dy}{dx} + \frac{2}{x}y = x$ , with factor $x^{2}$ .
$x^{2}y = \int x^{3}\,dx = \frac{x^{4}}{4} + C$ , so $y = \frac{x^{2}}{4} + \frac{C}{x^{2}}$ .

Worth noting

$\frac{x^{4}}{4} + C$ is $x^{2}y$ ; it still has to be divided by $x^{2}$ .

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

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