Which of the following function is homogeneous ?

Options

A. `f (x) =x sin y+y sin x`
B. `g(x)=xe^(y/x)+ye ^(x/y)`
C. `h (x)= (xy)/(x+y^(2))`
D. `phi (x) =(x-y cos x)/(y sin x+ y)`

Answer

To determine which of the given functions is homogeneous, we need to check each function by substituting \( x \) and \( y \) with \( tx \) and \( ty \) respectively, and see if the resulting expression can be factored into \( t^n \cdot f(x, y) \) for some integer \( n \).

### Step-by-Step Solution

1. **Identify the Functions**:
We have the following functions to analyze:
- \( f(x, y) = x \sin y + y \sin x \)
- \( g(x, y) = \frac{x e^y}{x + y} + \frac{y e^x}{y} \)
- \( h(x, y) = \frac{xy}{x + y^2} \)
- \( \phi(x, y) = x - y \cos x + y \sin x \)

2. **Check Function \( f(x, y) \)**:
- Substitute \( x \) and \( y \) with \( tx \) and \( ty \):
\[ f(tx, ty) = tx \sin(ty) + ty \sin(tx) \]
- Factor out \( t \):
\[ = t(x \sin(ty) + y \sin(tx)) \]
- Since \( \sin(ty) \) and \( \sin(tx) \) cannot be simplified to a form that allows factoring out \( t^n \), this function is **not homogeneous**.

3. **Check Function \( g(x, y) \)**:
- Substitute \( x \) and \( y \) with \( tx \) and \( ty \):
\[ g(tx, ty) = \frac{tx e^{ty}}{tx + ty} + \frac{ty e^{tx}}{ty} \]
- Factor out \( t \):
\[ = t \left( \frac{x e^{ty}}{x + y} + \frac{y e^{tx}}{y} \right) \]
- This can be expressed as \( t \cdot g(x, y) \), indicating that \( g(x, y) \) is **homogeneous**.

4. **Check Function \( h(x, y) \)**:
- Substitute \( x \) and \( y \) with \( tx \) and \( ty \):
\[ h(tx, ty) = \frac{(tx)(ty)}{tx + (ty)^2} \]
- Factor out \( t \):
\[ = \frac{t^2 xy}{t(x + t y^2)} = t \cdot \frac{xy}{x + ty^2} \]
- Since \( t \) remains in the denominator, this function is **not homogeneous**.

5. **Check Function \( \phi(x, y) \)**:
- Substitute \( x \) and \( y \) with \( tx \) and \( ty \):
\[ \phi(tx, ty) = tx - ty \cos(tx) + ty \sin(tx) \]
- Factor out \( t \):
\[ = t \left( x - y \cos(tx) + y \sin(tx) \right) \]
- Similar to \( f(x, y) \), the presence of \( \cos(tx) \) and \( \sin(tx) \) prevents it from being expressed as \( t^n \cdot \phi(x, y) \), thus this function is **not homogeneous**.

### Conclusion
The only homogeneous function among the given options is \( g(x, y) \).