The given equation is a partial differential equation (PDE):

$f_{xx} + x^2 f_{yy} = 0$

Here, $f_{xx}$ and $f_{yy}$ denote the second partial derivatives of the unknown function $f(x, y)$ with respect to $x$ and $y$, respectively.

Step-by-Step Approach:

To solve this equation, we can attempt methods like separation of variables or look for a specific type of solution (like assuming a solution in terms of a function of $x$ and $y$).

Step 1: Analyze the Equation

The given equation:

$f_{xx} + x^2 f_{yy} = 0$

is a linear second-order PDE with variable coefficients. It suggests the relationship between the curvature of the function $f(x, y)$ in the $x$-direction and the scaling of the curvature in the $y$-direction (scaled by $x^2$).

Step 2: Try Separation of Variables

A common technique for solving such PDEs is separation of variables, where we assume that the solution can be written as a product of functions of $x$ and $y$. So, we assume:

$f(x, y) = X(x) Y(y)$

Now, calculate the partial derivatives:

• $f_{xx} = X''(x) Y(y)$
• $f_{yy} = X(x) Y''(y)$

Substituting into the original PDE:

$X''(x) Y(y) + x^2 X(x) Y''(y) = 0$

Divide through by $X(x)Y(y)$:

$\frac{X''(x)}{X(x)} + x^2 \frac{Y''(y)}{Y(y)} = 0$

Step 3: Rearrange for Separation

For this equation to hold for all $x$ and $y$, each term must equal a constant, say $-\lambda$. Thus, we have two separate equations:

1. $\frac{X''(x)}{X(x)} = -\lambda$, which leads to the ODE for $X(x)$: $X''(x) + \lambda X(x) = 0$

2. $\frac{Y''(y)}{Y(y)} = \frac{\lambda}{x^2}$, which leads to the ODE for $Y(y)$: $Y''(y) + \frac{\lambda}{x^2} Y(y) = 0$

Step 4: Solve the ODEs

1. Solve for $X(x)$:

   The equation $X''(x) + \lambda X(x) = 0$ is a standard second-order linear ODE with constant coefficients. The solution depends on the sign of $\lambda$:

   • If $\lambda > 0$, the solution will involve trigonometric functions.
   • If $\lambda = 0$, the solution will be a linear function.
   • If $\lambda < 0$, the solution will involve exponential functions.

2. Solve for $Y(y)$:

   The equation $Y''(y) + \frac{\lambda}{x^2} Y(y) = 0$ may require further transformation or specific boundary conditions to solve.

Step 5: General Solution

The general solution of this PDE will involve combining the solutions to the two ODEs. The form will depend on the boundary conditions or any additional constraints provided.

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Would you like to explore more details about solving these ODEs? Let me know if you need further clarifications!

Here are some related questions you can explore:

1. What are common methods for solving partial differential equations with variable coefficients?
2. How does the method of separation of variables apply to higher-dimensional PDEs?
3. What is the role of boundary conditions in solving PDEs?
4. How can we use Fourier series to solve second-order linear PDEs?
5. What are some examples of real-world problems modeled by second-order PDEs?

Tip: Always check if the equation is homogeneous or inhomogeneous, as this significantly affects the solution approach.