The solution curve passes through (0, 1).

Question 57

HardOriginal question

Consider the differential equation $y\,e^{x/y}\,dx = (x\,e^{x/y} + y)\,dy$ , and put $x = vy$ .

The solution curve passes through $(0, 1)$ . What is $x$ when $y = e$ ?

0

Correct answerYour answer

\log 2

Correct answerYour answer

e \log 2

Correct answerYour answer

2 e

Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer C

Correct answer: C

At $(0, 1)$ : $e^{0} = \log 1 + C$ , so $C = 1$ .
At $y = e$ : $e^{x/e} = 1 + 1 = 2$ , so $\frac{x}{e} = \log 2$ and $x = e\log 2$ .

Worth noting

$\log 2$ is $v = \frac{x}{y}$ ; $x$ is $v$ times $y = e$ .

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

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