For $f(x, y) = y^{2}e^{x/y} + xy$ , which of the following statements are true?
I. $f$ is homogeneous of degree 2.
II. $f(3x, 3y) = 9f(x, y)$ .
III. $f(x, y) = y^{2}g\left(\frac{x}{y}\right)$ with $g(u) = e^{u} + u$ .
IV. $\frac{dy}{dx} = f(x, y)$ is a homogeneous differential equation.

All four: I, II, III and IVCorrect answerYour answerI and II onlyCorrect answerYour answerI and III onlyCorrect answerYour answerI, II and III onlyCorrect answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer D

Correct answer: D

$f(\lambda x, \lambda y) = \lambda^{2}y^{2}e^{x/y} + \lambda^{2}xy = \lambda^{2}f$ : I holds, II is the case $\lambda = 3$ , and dividing by $y^{2}$ gives III.
IV is false: $\frac{dy}{dx} = F$ is homogeneous only when $F$ has degree 0, and $f$ has degree 2.

Worth noting

IV mixes up a homogeneous function with a homogeneous equation, which needs degree 0.

Concept: Facts about one homogeneous function
Typical timeTypical: 150 s
Your timeYours: —

The rule this testsHomogeneous equations

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