What is the general solution of dx/dy - 2x = e^y?

Homogeneous & Linear Differential Equations · Question 107 of 120

Question 107

MediumOriginal question

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

y = Ce^{2x} - e^{x}

Correct answerYour answer

x = Ce^{2y} + e^{y}

Correct answerYour answer

x = Ce^{2y} - e^{y}

Correct answerYour answer

x = Ce^{-2y} - e^{y}

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Show the answer and explanation

CorrectNot quiteAnswer C

Correct answer: C

Linear in $x$ with $P_{1} = -2$ : the factor is $e^{-2y}$ , and $\frac{d}{dy}\left(xe^{-2y}\right) = e^{-y}$ .
$xe^{-2y} = -e^{-y} + C$ , so $x = Ce^{2y} - e^{y}$ .

Worth noting

$y = Ce^{2x} - e^{x}$ swaps the roles of $x$ and $y$ ; here $x$ is the unknown.

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

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