The problem asks to find the second solution $y_2(x)$ of the differential equation using the reduction of order method. The given equation is:

$y'' - 12y' + 36y = 0$

The first solution $y_1(x) = e^{6x}$ is provided.

To use the reduction of order formula, we are given the following general form for the second solution $y_2(x)$:

$y_2(x) = y_1(x) \int \frac{e^{-\int P(x) dx}}{(y_1(x))^2} dx$

Step-by-Step Approach:

1. Rewrite the given differential equation in standard form: $y'' - 12y' + 36y = 0$ In comparison to the general second-order linear equation $y'' + P(x)y' + Q(x)y = 0$, we see: $P(x) = -12 \quad \text{and} \quad Q(x) = 36.$

2. Use the reduction of order formula:

   • The given solution is $y_1(x) = e^{6x}$.
   • $P(x) = -12$, so $\int P(x) dx = \int (-12) dx = -12x$.

3. Set up the formula for the second solution: $y_2(x) = y_1(x) \int \frac{e^{-\int P(x) dx}}{(y_1(x))^2} dx$ Substitute the known values: $y_2(x) = e^{6x} \int \frac{e^{-(-12x)}}{(e^{6x})^2} dx$ Simplify: $y_2(x) = e^{6x} \int \frac{e^{12x}}{e^{12x}} dx = e^{6x} \int 1 dx$ $y_2(x) = e^{6x} \cdot x + C.$

Thus, the second solution is:

$y_2(x) = x e^{6x}.$

Would you like further explanation on any step or more details? Here are some related questions to explore:

1. What is the significance of the reduction of order method?
2. Why do we need two independent solutions for second-order differential equations?
3. How does the characteristic equation help solve homogeneous linear differential equations?
4. What is the Wronskian, and how can it confirm the independence of solutions?
5. Can you use variation of parameters to solve non-homogeneous differential equations?

Tip: Always check if the two solutions you find are linearly independent to ensure they form a general solution to the differential equation.