Which differential equation does the family y = A/x + B satisfy, A, B being arbitrary constants?

Question 41

MediumOriginal question

Which differential equation does the family $y = \frac{A}{x} + B$ satisfy, $A, B$ being arbitrary constants?

x\frac{d^2y}{dx^2} - 2\frac{dy}{dx} = 0

Correct answerYour answer

x\frac{dy}{dx} + y = 0

Correct answerYour answer

\frac{d^2y}{dx^2} = 0

Correct answerYour answer

x\frac{d^2y}{dx^2} + 2\frac{dy}{dx} = 0

Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer D

Correct answer: D

$\frac{dy}{dx} = -\frac{A}{x^2}$ , so $x^2\frac{dy}{dx} = -A$ .
Differentiate again: $2x\frac{dy}{dx} + x^2\frac{d^2y}{dx^2} = 0$ , that is $x\frac{d^2y}{dx^2} + 2\frac{dy}{dx} = 0$ .

Worth noting

$x\frac{dy}{dx} + y = 0$ is the equation of $y = \frac{A}{x}$ alone; the $+B$ needs a second derivative to remove.

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