What is the general solution of dx/dy + x = y?

Homogeneous & Linear Differential Equations · Question 111 of 120

Question 111

HardOriginal question

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

y = x - 1 + Ce^{-x}

Correct answerYour answer

x = y + 1 + Ce^{-y}

Correct answerYour answer

x = y - 1 + Ce^{-y}

Correct answerYour answer

x = y - 1 + Ce^{y}

Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer C

Correct answer: C

Linear in $x$ with $P = 1$ : factor $e^{y}$ , and $\frac{d}{dy}(xe^{y}) = ye^{y}$ .
$xe^{y} = (y - 1)e^{y} + C$ , so $x = y - 1 + Ce^{-y}$ .

Worth noting

$y = x - 1 + Ce^{-x}$ 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.