The homogeneous function $F(x, y) = x^{3} + 2x^{2}y - y^{3}$ is written as $x^{3}h(v)$ , where $v = \frac{y}{x}$ . What is $h(v)$ ?

1 + 2v - v^{3}

Correct answerYour answer

v^{3} + 2v^{2} - 1

Correct answerYour answer

1 + 2v^{2} - v^{3}

Correct answerYour answer

1 + 2v - v^{2}

Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer A

Correct answer: A

Divide each term by $x^{3}$ : $\frac{2x^{2}y}{x^{3}} = 2\frac{y}{x}$ and $\frac{y^{3}}{x^{3}} = \left(\frac{y}{x}\right)^{3}$ .
So $h(v) = 1 + 2v - v^{3}$ .

Worth noting

$v^{3} + 2v^{2} - 1$ is $F$ divided by $y^{3}$ , a function of $\frac{x}{y}$ , not of $v = \frac{y}{x}$ .

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

The homogeneous function F(x,y)=x3+2x2y−y3F(x, y) = x^{3} + 2x^{2}y - y^{3}F(x,y)=x3+2x2y−y3 is written as x3h(v)x^{3}h(v)x3h(v), where v=yxv = \frac{y}{x}v=xy​. What is h(v)h(v)h(v)?