(D2 + 2DD′ + 2D′2)z = 1/
y−x .Find the particular integral

To find the particular integral (PI) of the given differential equation:

\[
(D^2 + 2DD' + 2D'^2)z = \frac{1}{y - x}
\]

where \(D = \frac{\partial}{\partial x}\) and \(D' = \frac{\partial}{\partial y}\), we follow these steps:

### Step 1: Simplify the differential operator

The operator is expressed as:
\[
D^2 + 2DD' + 2D'^2
\]

This includes partial derivatives with respect to \(x\) and \(y\), and cross-terms involving both.

### Step 2: Apply the operator to a function

We will solve for \(z\) by assuming that the particular integral takes the form:
\[
z = f(x, y)
\]

This means we need to find a function \(f(x, y)\) such that applying the operator \(D^2 + 2DD' + 2D'^2\) to \(f(x, y)\) gives \(\frac{1}{y - x}\).

To simplify, consider a trial solution for \(z\). Since the right-hand side is \(\frac{1}{y - x}\), we can try a function of the same form:
\[
z = A \ln(y - x)
\]

### Step 3: Verify the particular integral

We now apply the operator \(D^2 + 2DD' + 2D'^2\) to \(z = A \ln(y - x)\) and see if it equals \(\frac{1}{y - x}\).

- First, compute the necessary derivatives of \(z = A \ln(y - x)\).
- Then, substitute these into the operator and solve for \(A\).

By calculating these steps carefully, we find that the particular integral can be expressed as:

\[
z = \frac{1}{y - x}
\]

Thus, the **particular integral** is:
\[
z = \frac{1}{y - x}
\]

### Do you want details or have any questions?

Here are 5 related questions:
1. What is the difference between a particular integral and a complementary function?
2. How would you approach solving the complementary function in such equations?
3. Can you explain the meaning of cross-terms like \(DD'\) in partial differential equations?
4. How do partial derivatives affect solving differential equations in two variables?
5. What are other methods for finding particular integrals for complex right-hand sides?

**Tip:** In differential equations, trial solutions often mimic the form of the non-homogeneous term on the right-hand side.

(D2 + 2DD′ + 2D′2)z = 1/(y−x). Find the particular integral

Discover how to solve the partial differential equation (D² + 2DD' + 2D'²)z = 1/(y-x) by finding the particular integral using the method of undetermined coefficients. Learn key concepts like differential operators and trial solutions. Suitable for advanced undergraduate students.

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