1. Tardigrade
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  4. A function F(x, y) is a homogeneous function of degree n, if F(x, y)=xn g((y/x)) or yn h((x/y)) On the basis of above information, which of the following function is homogeneous function of degree 0 ?

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Q. A function $F(x, y)$ is a homogeneous function of degree $n$, if $F(x, y)=x^n g\left(\frac{y}{x}\right)$ or $y^n h\left(\frac{x}{y}\right)$
On the basis of above information, which of the following function is homogeneous function of degree 0 ?

Differential Equations

Solution:

A function $F(x, y)$ is a homogeneous function of degree $n$ if
$F(x, y)=x^n g\left(\frac{y}{x}\right) \text { or } y^n h\left(\frac{x}{y}\right)$
We also observe that
$F_1(x, y)=x^2\left(\frac{y^2}{x^2}+\frac{2 y}{x}\right)=x^2 h_1\left(\frac{y}{x}\right)$
or $ F_1(x, y)=y^2\left(1+\frac{2 x}{y}\right)=y^2 h_2\left(\frac{x}{y}\right)$
$F_2(x, y)=x\left(2-\frac{3 y}{x}\right)=x h_3\left(\frac{y}{x}\right)$
or $ F_2(x, y)=y\left(2 \frac{x}{y}-3\right)=y h_4\left(\frac{x}{y}\right) $
$ F_3(x, y)=x^0 \cos \left(\frac{y}{x}\right)=x^0 h_5\left(\frac{y}{x}\right) $
$ F_4(x, y) \neq x^n h_6\left(\frac{y}{x}\right)$
for any $n \in W$
or $ F_4(x, y) \neq y^n h_7\left(\frac{x}{y}\right), $ for any $n \in W$
$\because F_3(x, y)=x^0 h_5\left(\frac{y}{x}\right)$
$\Rightarrow F_3(x, y)$ is a homogeneous function of degree 0 .