Question Statement

Determine whether the function $f (x, y) = x^{2} - y$ is a homogeneous function. If it is homogeneous, identify its degree; if not, show why the homogeneity condition fails.

Background and Explanation

A function $f (x, y)$ is homogeneous of degree $n$ if scaling both variables by a factor $t$ results in the function being scaled by $t^{n}$ , i.e., $f (t x, t y) = t^{n} f (x, y)$ for some constant $n$ . This property requires all terms in the function to have the same total degree.

Solution

To test for homogeneity, we substitute $x \to t x$ and $y \to t y$ into the function and check if the result can be written as $t^{n} f (x, y)$ for some constant $n$ .

Starting with the given function: $f (x, y) = x^{2} - y$

Replace $x$ by $t x$ and $y$ by $t y$ : $f (t x, t y) = (t x)^{2} - (t y) = t^{2} x^{2} - t y = t^{2} (x^{2} - \frac{y}{t})$

For the function to be homogeneous of degree 2, we would need $f (t x, t y) = t^{2} f (x, y) = t^{2} (x^{2} - y) = t^{2} x^{2} - t^{2} y$ . However, comparing this with our result: $t^{2} x^{2} - t y \neq = t^{2} x^{2} - t^{2} y$

Since the second term contains $t y$ (degree 1 in $t$ ) rather than $t^{2} y$ (degree 2 in $t$ ), we cannot factor out a consistent power of $t$ from all terms. Therefore: $f (t x, t y) \neq = t^{n} f (x, y) for any constant n$

Conclusion: $f (x, y)$ is not a homogeneous function.

Key Formulas or Methods Used

Definition of homogeneous function: $f (t x, t y) = t^{n} f (x, y)$ for some degree $n$
Degree of a term: The sum of the exponents of variables in that term (e.g., $x^{2}$ has degree 2, $y$ has degree 1)
Substitution method: Replacing variables with scaled versions $(t x, t y)$ to test functional properties

Summary of Steps

State the function: Begin with $f (x, y) = x^{2} - y$ .
Apply scaling: Substitute $x \to t x$ and $y \to t y$ to obtain $f (t x, t y) = (t x)^{2} - t y$ .
Simplify: Expand to get $t^{2} x^{2} - t y$ .
Analyze degrees: Observe that the first term scales as $t^{2}$ while the second scales as $t^{1}$ .
Conclude: Since different terms scale with different powers of $t$ , the function cannot be written as $t^{n} f (x, y)$ , proving it is not homogeneous.

Solution

To test for homogeneity, we substitute

x \to t x

and

y \to t y

into the function and check if the result can be written as

t^{n} f (x, y)

for some constant

n

Starting with the given function:

f (x, y) = x^{2} - y

Replace

x

t x

and

y

t y

f (t x, t y) = (t x)^{2} - (t y) = t^{2} x^{2} - t y = t^{2} (x^{2} - \frac{y}{t})

For the function to be homogeneous of degree 2, we would need

f (t x, t y) = t^{2} f (x, y) = t^{2} (x^{2} - y) = t^{2} x^{2} - t^{2} y

. However, comparing this with our result:

t^{2} x^{2} - t y \neq = t^{2} x^{2} - t^{2} y

Since the second term contains

t y

(degree 1 in

t

) rather than

t^{2} y

(degree 2 in

t

), we cannot factor out a consistent power of

t

from all terms. Therefore:

f (t x, t y) \neq = t^{n} f (x, y) for any constant n

Conclusion:

f (x, y)

is not a homogeneous function.

Summary of Steps

State the function: Begin with

f (x, y) = x^{2} - y

Apply scaling: Substitute

x \to t x

and

y \to t y

to obtain

f (t x, t y) = (t x)^{2} - t y

Simplify: Expand to get

t^{2} x^{2} - t y

Analyze degrees: Observe that the first term scales as

t^{2}

while the second scales as

t^{1}

Conclude: Since different terms scale with different powers of

t

, the function cannot be written as

t^{n} f (x, y)

, proving it is not homogeneous.

4.3 Q-2

Question Statement

Background and Explanation

Solution

Key Formulas or Methods Used

Summary of Steps

4.3 Q-2

Question Statement

Background and Explanation

Solution

Key Formulas or Methods Used

Summary of Steps