To solve this problem using Euler’s theorem for homogeneous functions, we need to understand the structure of the function $z = x^n f_1\left(\frac{y}{x}\right) + y^n f_2\left(\frac{x}{y}\right)$ and confirm that it satisfies the given partial differential equation.

Step-by-Step Solution Outline

1. Identify the Homogeneous Terms:

   • Notice that $x^n f_1\left(\frac{y}{x}\right)$ and $y^n f_2\left(\frac{x}{y}\right)$ suggest that $z$ is a homogeneous function of $x$ and $y$ with degree $n$.

2. Apply Euler’s Theorem on Homogeneous Functions:

   • For a homogeneous function $z(x, y)$ of degree $n$, Euler’s theorem tells us: $x \frac{\partial z}{\partial x} + y \frac{\partial z}{\partial y} = n z$

3. Second-order Derivatives:

   • To satisfy the equation: $x^2 \frac{\partial^2 z}{\partial x^2} + 2xy \frac{\partial^2 z}{\partial x \partial y} + y^2 \frac{\partial^2 z}{\partial y^2} + x \frac{\partial z}{\partial x} + y \frac{\partial z}{\partial y} = n^2 z,$ we need to calculate the second partial derivatives of $z$ and plug them into this expression.

4. Verification:

   • By calculating each term individually and substituting into the equation, we can verify if the expression simplifies to $n^2 z$.

Solution Outline Explanation:

1. Calculate $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$:

   • Differentiate $z = x^n f_1\left(\frac{y}{x}\right) + y^n f_2\left(\frac{x}{y}\right)$ with respect to $x$ and $y$.

2. Calculate $\frac{\partial^2 z}{\partial x^2}$, $\frac{\partial^2 z}{\partial x \partial y}$, and $\frac{\partial^2 z}{\partial y^2}$:

   • Take the second derivatives as needed and plug them back into the equation to confirm it results in $n^2 z$.

Would you like to see each step in detail?

Here are some follow-up questions for further exploration:

1. What is Euler’s theorem, and how does it apply to homogeneous functions?
2. How do we identify the degree of homogeneity in a function?
3. How do we handle partial derivatives for functions with mixed terms like $x^n f_1\left(\frac{y}{x}\right)$?
4. Why does the equation reduce to $n^2 z$ when Euler’s theorem is applied?
5. How can similar problems be solved using other differential operators?

Tip: When working with homogeneous functions, check if the function’s degree can simplify your differentiation steps, as Euler's theorem can often reduce the complexity.