Solve the initial value problem. $$ \begin{array}{l}\frac{d y}{d x}=18(6 x-5)^{-6}, y(0)=1 \ y=-\frac{18}{5}(6 x-5)^{-5}+1 \ y=-\frac{3}{5}(6 x-5)^{-5}+1 \ y=-\frac{3}{7}(6 x-5)^{-7}+1 \ y=(6 x-5)^{-5}-5\end{array} $$

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To solve the initial value problem given by the differential equation $$ \frac{d y}{d x}=18(6 x-5)^{-6} $$ with the initial condition $$ y(0)=1 $$, we will integrate the right-hand side and then apply the initial condition to find the constant of integration.

1. **Integrate the right-hand side:**
   $$
   y = \int 18(6 x - 5)^{-6} \, dx
   $$

2. **Use substitution for integration:**
   Let $$ u = 6x - 5 $$, then $$ \frac{du}{dx} = 6 $$ or $$ dx = \frac{du}{6} $$.

The integral becomes:
   $$
   y = \int 18 u^{-6} \cdot \frac{du}{6} = 3 \int u^{-6} \, du
   $$

3. **Integrate $$ u^{-6} $$:**
   $$
   \int u^{-6} \, du = -\frac{1}{5} u^{-5} + C
   $$
   Therefore,
   $$
   y = 3 \left(-\frac{1}{5} u^{-5}\right) + C = -\frac{3}{5} u^{-5} + C
   $$

4. **Substituting back for $$ u $$:**
   $$
   y = -\frac{3}{5} (6x - 5)^{-5} + C
   $$

5. **Apply the initial condition $$ y(0) = 1 $$:**
   $$
   1 = -\frac{3}{5} (6(0) - 5)^{-5} + C
   $$
   $$
   1 = -\frac{3}{5} (-5)^{-5} + C
   $$
   $$
   1 = -\frac{3}{5} \left(-\frac{1}{3125}\right) + C
   $$
   $$
   1 = \frac{3}{15625} + C
   $$
   $$
   C = 1 - \frac{3}{15625}
   $$

6. **Calculate $$ C $$:**
   $$
   C = \frac{15625 - 3}{15625} = \frac{15622}{15625}
   $$

7. **Final solution:**
   $$
   y = -\frac{3}{5} (6x - 5)^{-5} + \frac{15622}{15625}
   $$

Now, let's check if any of the provided options match this solution. The options given are:

1. $$ y = -\frac{18}{5}(6 x-5)^{-5}+1 $$
2. $$ y = -\frac{3}{5}(6 x-5)^{-5}+1 $$
3. $$ y = -\frac{3}{7}(6 x-5)^{-7}+1 $$
4. $$ y = (6 x-5)^{-5}-5 $$

We can see that the second option $$ y = -\frac{3}{5}(6 x-5)^{-5}+1 $$ can be rewritten as:
$$
y = -\frac{3}{5}(6 x-5)^{-5} + 1
$$
This matches our derived solution, confirming that the correct answer is:

**Answer:** $$ y = -\frac{3}{5}(6 x-5)^{-5}+1 $$
The solution to the initial value problem is $$ y = -\frac{3}{5}(6x - 5)^{-5} + 1 $$.

Solve the initial value problem.

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