Let's first analyze the given equation, which seems to be a second-order partial differential equation (PDE):

$\frac{\partial^2}{\partial x^2} - 7 \frac{\partial}{\partial x} + 12 \frac{\partial^2}{\partial y^2} = e^{x-y}$

Here’s the breakdown:

1. Left-hand side terms: The terms involve partial derivatives:

   • $\frac{\partial^2}{\partial x^2}$: Second derivative with respect to $x$.
   • $\frac{\partial}{\partial x}$: First derivative with respect to $x$.
   • $\frac{\partial^2}{\partial y^2}$: Second derivative with respect to $y$.

2. Right-hand side: It is simply an exponential function of $x$ and $y$, namely $e^{x-y}$.

Solution Strategy

To solve this PDE, we might consider separation of variables or use specific methods like the method of undetermined coefficients or trial solutions for specific forms (e.g., exponential functions). Given that the right-hand side is $e^{x-y}$, we might look for a solution of the form $u(x, y) = A e^{x-y}$.

Let’s proceed by testing this form of solution.

Step 1: Substituting the trial solution

Let $u(x, y) = A e^{x-y}$, where $A$ is a constant to be determined.

Now, we calculate the required derivatives:

1. First derivatives: $\frac{\partial u}{\partial x} = A e^{x-y}$ $\frac{\partial u}{\partial y} = -A e^{x-y}$

2. Second derivatives: $\frac{\partial^2 u}{\partial x^2} = A e^{x-y}$ $\frac{\partial^2 u}{\partial y^2} = A e^{x-y}$

Step 2: Substituting into the PDE

Now, substitute these derivatives into the original PDE:

$\frac{\partial^2 u}{\partial x^2} - 7 \frac{\partial u}{\partial x} + 12 \frac{\partial^2 u}{\partial y^2} = e^{x-y}$

This becomes:

$A e^{x-y} - 7 A e^{x-y} + 12 A e^{x-y} = e^{x-y}$

Simplify the left-hand side:

$(1 - 7 + 12) A e^{x-y} = e^{x-y}$

$6 A e^{x-y} = e^{x-y}$

Step 3: Solving for $A$

To satisfy this equation, we must have $6A = 1$, so:

$A = \frac{1}{6}$

Thus, the solution to the PDE is:

$u(x, y) = \frac{1}{6} e^{x-y}$

Conclusion

The solution to the given PDE is:

$u(x, y) = \frac{1}{6} e^{x-y}$

Let me know if you need any more details or clarification!

Related Questions:

1. What are the general methods for solving second-order linear partial differential equations?
2. How do separation of variables work in the context of PDEs?
3. What is the role of trial solutions in solving PDEs?
4. How would you solve a PDE with a non-exponential right-hand side function?
5. How can boundary conditions influence the solution of a PDE?

Tip: When solving PDEs, always check for the form of the solution based on the right-hand side. For exponential functions, trial solutions with similar forms are often very effective.