Read the two statements and choose the correct option. I. For dx/dy = x/y + tanx/y, the substitution x = vy gives cot v dv = dy/y. II. The general solution of that equation is sinx/y = Cy.

Question 49

MediumOriginal question

Read the two statements and choose the correct option.
I. For $\frac{dx}{dy} = \frac{x}{y} + \tan\frac{x}{y}$ , the substitution $x = vy$ gives $\cot v\,dv = \frac{dy}{y}$ .
II. The general solution of that equation is $\sin\frac{x}{y} = Cy$ .

Only ICorrect answerYour answerOnly IICorrect answerYour answerBoth I and IICorrect answerYour answerNeither I nor IICorrect answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer C

Correct answer: C

I: $v + y\frac{dv}{dy} = v + \tan v$ , so $\frac{dv}{\tan v} = \cot v\,dv = \frac{dy}{y}$ . True.
II: $\log\lvert \sin v \rvert = \log\lvert y \rvert + c$ , so $\sin\frac{x}{y} = Cy$ . True.

Worth noting

With $x = vy$ the constant multiplies $y$ , the independent variable, not $x$ .

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