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### (a) Show that \(\frac{xy}{x^2 + y^2}\) is a homogeneous function in \(x\) and \(y\).

A function \(f(x, y)\) is homogeneous of degree \(n\) if for some real number \(n\), we have:
\[
f(tx, ty) = t^n f(x, y)
\]
for all \(t\). Let's test this for the given function \(f(x, y) = \frac{xy}{x^2 + y^2}\).

Substitute \(x \to tx\) and \(y \to ty\):
\[
f(tx, ty) = \frac{(tx)(ty)}{(tx)^2 + (ty)^2} = \frac{t^2xy}{t^2(x^2 + y^2)} = \frac{t^2xy}{t^2(x^2 + y^2)} = \frac{xy}{x^2 + y^2}.
\]
Since the result is the same as the original function, we conclude that:
\[
f(x, y) = \frac{xy}{x^2 + y^2}
\]
is homogeneous of degree 0 (since \(t^0 = 1\)).

### (b) Transform the equation using \(y = vx\).

Given the differential equation:
\[
\frac{dy}{dx} = \frac{xy}{x^2 + y^2},
\]
we substitute \(y = vx\), where \(v\) is a function of \(x\). Thus:
\[
\frac{dy}{dx} = v + x \frac{dv}{dx}.
\]
Substitute \(y = vx\) into the right-hand side of the equation:
\[
\frac{dy}{dx} = \frac{x(vx)}{x^2 + (vx)^2} = \frac{x^2v}{x^2 + x^2v^2} = \frac{v}{1 + v^2}.
\]
Now, the equation becomes:
\[
v + x \frac{dv}{dx} = \frac{v}{1 + v^2}.
\]

### (c) Solve the resulting equation using separation of variables.

Rearranging the equation:
\[
x \frac{dv}{dx} = \frac{v}{1 + v^2} - v = \frac{v(1 - v^2)}{1 + v^2}.
\]
Now, separate the variables:
\[
\frac{1 + v^2}{v(1 - v^2)} dv = \frac{dx}{x}.
\]
Integrating both sides:
\[
\int \frac{1 + v^2}{v(1 - v^2)} dv = \int \frac{dx}{x}.
\]

The integral on the left-hand side can be simplified and solved using partial fractions. Would you like further detailed steps for this integration or have any questions on this part?

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Here are five questions to further explore this problem:
1. What other techniques can you use to solve homogeneous differential equations?
2. How can the degree of homogeneity help simplify differential equations?
3. What are the advantages of using the substitution \(y = vx\) in differential equations?
4. Can this equation be solved without separation of variables?
5. How would you verify the solution once found?

**Tip:** Always verify if an equation is homogeneous before applying substitutions like \(y = vx\), as this simplifies solving the equation.

Show that (xy / (x^2 + y^2)) is a homogeneous function in x and y. Using the substitution y = vx transform the equation (dy/dx = xy / (x^2 + y^2)) into an equation containing v and x only. Hence solve the resulting equation using the method of separation of variables.

This problem shows how to determine if a function is homogeneous, then uses substitution to transform a differential equation and solve it using separation of variables. Key concepts include homogeneous functions, differential equations, and substitution methods.

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