To solve the initial value problem given by the differential equation $\sqrt{y} \, dx + (x - 5) \, dy = 0$ with the initial condition $y(6) = 9$, follow these steps:

1. Separate the variables: Rewrite the equation to separate $x$ and $y$.

   $\sqrt{y} \, dx + (x - 5) \, dy = 0$ Divide through by $(x - 5) \sqrt{y}$:

   $\frac{dx}{x - 5} + \frac{dy}{\sqrt{y}} = 0$

2. Integrate both sides: Integrate each term separately.

   $\int \frac{dx}{x - 5} + \int \frac{dy}{\sqrt{y}} = 0$

   $\int \frac{dx}{x - 5} = \ln|x - 5|$

   $\int \frac{dy}{\sqrt{y}} = 2\sqrt{y}$

   So we have:

   $\ln|x - 5| + 2\sqrt{y} = C$

3. Solve for the constant $C$ using the initial condition $y(6) = 9$:

   $\ln|6 - 5| + 2\sqrt{9} = C$

   $\ln 1 + 2 \times 3 = C$

   $0 + 6 = C$

   $C = 6$

4. Form the general solution using the constant found:

   $\ln|x - 5| + 2\sqrt{y} = 6$

Therefore, the implicit solution to the initial value problem is:

$\ln|x - 5| + 2\sqrt{y} = 6$

Would you like any further details or explanations?

Here are some more questions you might find useful to explore:

1. How do you solve separable differential equations in general?
2. What are the methods for solving linear differential equations?
3. How do you interpret the initial conditions in differential equations?
4. Can you provide examples of other types of differential equations?
5. How do you verify the solution to a differential equation?

Tip: Always check your solution by substituting it back into the original differential equation to ensure it satisfies both the equation and the initial conditions.