Question

Homogeneous Differential:
Determine whether the given function is homogeneous. If so, state the degree of homogeneity.
\[cos\frac{x^2}{x+y}\]
The book says that this is not homogeneous, but I do not understand why or how to verify this.

Answer 1

Is there a differential somewhere?

Answer 2

No, the only thing required is to determine if this equation is homogeneous.

Answer 3

i believe theres some book work that says; if f(tx,ty) = t^n f(x,y) then its homogenous, if not; then it aint

Answer 4

Can it simply be that the numerator is a power of 2 and the denominator has powers of 1?

Answer 5

if\[cos\frac{(tx)^2}{tx+ty}=t^n~cos\frac{x^2}{x+y}\]it would be homogenous

Answer 6

Yes, but when you work it out you get \[\frac{t^2}{t} cos\frac{x^2}{x+y}\]

Answer 7

By that logic you would think it homogeneous, yet the book says it is not.

Answer 8

can you show the steps that allows you to pull out a t?

Answer 9

@amistre64 I used essentially the same method putting them in. \[cos\frac{(tx)^2}{t(x+y)} \rightarrow cos\frac{t^2x^2}{tx+ty} \rightarrow cos\frac{t^2}{t}\frac{x^2}{x+y}\]

Answer 10

have you seen the series expansion for this by chance?

Answer 11

Would this be \[cos\frac{t^2}{t} \times cos\frac{x^2}{x+y}\]?

I have not or do not recall the series expansion.

Answer 12

\[cos(t\frac{x^2}{x+y})\ne t^ncos(\frac{x^2}{x+y})\]
or am i missing something?

Answer 13

@amistre64 That makes sense, it seems it was I that was missing it. :) Thanks so much.

Answer 14

the series expansion shows something to the effect of\[1-\frac{t^2x^4}{2y^2}+\frac{t^2x^5}{y^3}...\]
which of course cant factor out the t^2 wither

Answer 15

good luck ;)