The solution curve passes through (1, 0).

Question 53

HardOriginal question

Consider the differential equation $x^{2}\frac{dy}{dx} = x^{2} + xy + y^{2}$ , and put $y = vx$ .

The solution curve passes through $(1, 0)$ . What is $y$ when $x = e^{\pi/4}$ ?

\frac{\pi}{4}

Correct answerYour answer

1

Correct answerYour answer

\frac{\pi e^{\frac{\pi}{4}}}{4}

Correct answerYour answer

e^{\frac{\pi}{4}}

Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer D

Correct answer: D

At $(1, 0)$ : $\tan^{-1}0 = \log 1 + C$ , so $C = 0$ and $\tan^{-1}\frac{y}{x} = \log x$ .
At $x = e^{\pi/4}$ : $\tan^{-1}\frac{y}{x} = \frac{\pi}{4}$ , so $\frac{y}{x} = 1$ and $y = e^{\pi/4}$ .

Worth noting

1 is $v = \frac{y}{x}$ ; $y$ is $v$ times $x$ .

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

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