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Homogeneous & Linear Differential Equations · Question 56 of 120

What is the general solution of the equation?

Question 56

HardOriginal question

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

What is the general solution of the equation?

\tan^{-1}\frac{y}{x} = \log\lvert x \rvert + C

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\tan^{-1}\frac{x}{y} = \log\lvert x \rvert + C

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\log\left(1 + \frac{y^{2}}{x^{2}}\right) = \log\lvert x \rvert + C

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\tan^{-1}\frac{y}{x} = \log\lvert y \rvert + C

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Show the answer and explanation

CorrectNot quiteAnswer A

Correct answer: A

$\int \frac{dv}{1 + v^{2}} = \tan^{-1}v$ and $\int \frac{dx}{x} = \log\lvert x \rvert$ .
$\tan^{-1}\frac{y}{x} = \log\lvert x \rvert + C$ .

Worth noting

$\log\left(1 + v^{2}\right)$ integrates $\frac{1}{1 + v^{2}}$ as if it were $\frac{2v}{1 + v^{2}}$ .

Concept: Solving a homogeneous equation
Typical timeTypical: 150 s
Your timeYours: —

The rule this testsSolving homogeneous equations

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