Which of these equations is linear when $x$ is taken as the unknown, a function of $y$ ?

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

Correct answerYour answer

\frac{dy}{dx} = \frac{1}{x + y^{2}}

Correct answerYour answer

\frac{dy}{dx} = \frac{1}{x^{2} + y}

Correct answerYour answer

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

Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer B

Correct answer: B

Flip each one to $\frac{dx}{dy}$ : $\frac{dy}{dx} = \frac{1}{x + y^{2}}$ gives $\frac{dx}{dy} - x = y^{2}$ , linear in $x$ with $P_{1} = -1$ .
The others give $\frac{dx}{dy} = \frac{1}{x^{2} - y}$ , $x^{2} + y$ and $\frac{x^{2} + y}{y}$ , each with $x^{2}$ in it.

Worth noting

$\frac{dy}{dx} + y = x^{2}$ is linear in $y$ ; flipped, $\frac{dx}{dy} = \frac{1}{x^{2} - y}$ is not linear in $x$ .

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

Which of these equations is linear when xxx is taken as the unknown, a function of yyy?