We are given the equation:

$\frac{a}{b} - \frac{b}{a} = \frac{a+b}{ab}$

Step 1: Simplify the left-hand side (LHS)

The left-hand side is the difference between two fractions with different denominators:

$\frac{a}{b} - \frac{b}{a}$

To combine them, we need a common denominator, which is $ab$:

$\frac{a}{b} - \frac{b}{a} = \frac{a^2}{ab} - \frac{b^2}{ab} = \frac{a^2 - b^2}{ab}$

Thus, the left-hand side simplifies to:

$\frac{a^2 - b^2}{ab}$

Step 2: Set the simplified LHS equal to the right-hand side (RHS)

Now the equation becomes:

$\frac{a^2 - b^2}{ab} = \frac{a + b}{ab}$

Step 3: Eliminate the denominators

Both sides have the denominator $ab$, so we can cancel it out:

$a^2 - b^2 = a + b$

Step 4: Factor the left-hand side

The left-hand side is a difference of squares, which can be factored:

$(a - b)(a + b) = a + b$

Step 5: Solve for $a$ and $b$

We now have:

$(a - b)(a + b) = a + b$

There are two possibilities:

1. $a + b = 0$, which gives the solution $a = -b$.
2. If $a + b \neq 0$, we can divide both sides by $a + b$:

$a - b = 1$

Step 6: Solve the two cases

Case 1: $a = -b$

Substituting this into the original equation:

$a + b = 0$

This is one set of solutions where $a = -b$.

Case 2: $a - b = 1$

From this, we get:

$a = b + 1$

Final Answer:

The solutions to the equation are:

$a = -b \quad \text{or} \quad a = b + 1$

Would you like further details on this?

Here are 5 questions related to this topic:

1. How can the method of factoring be applied to other types of equations?
2. What is the significance of the difference of squares in solving equations?
3. How do we handle equations when variables appear in both the numerator and denominator?
4. Can you solve an equation where both sides are more complex rational expressions?
5. What if the equation contained higher powers, like $a^3$ or $b^3$?

Tip: Always check if you can factor expressions in equations before moving on to more complex methods!