Find the function​ y(x) satisfying StartFraction dy Over dx EndFraction equals 3 x Superscript negative 6 divided by 7
 and ​y(minus​1)equalsnegative 8.

To find the function \( y(x) \) satisfying the given differential equation \(\frac{dy}{dx} = \frac{3x^{-6}}{7}\) and the initial condition \( y(-1) = -8 \), we need to perform the following steps:

1. **Solve the differential equation:**

   \[
   \frac{dy}{dx} = \frac{3}{7} x^{-6}
   \]

   Integrate both sides with respect to \( x \):

   \[
   y = \int \frac{3}{7} x^{-6} \, dx
   \]

   \[
   y = \frac{3}{7} \int x^{-6} \, dx
   \]

   Recall that the integral of \( x^n \) is \(\frac{x^{n+1}}{n+1} \) (for \( n \neq -1 \)):

   \[
   y = \frac{3}{7} \cdot \frac{x^{-5}}{-5}
   \]

   Simplify the constant factor:

   \[
   y = -\frac{3}{35} x^{-5} + C
   \]

   So the general solution is:

   \[
   y = -\frac{3}{35} x^{-5} + C
   \]

2. **Apply the initial condition \( y(-1) = -8 \):**

   \[
   -8 = -\frac{3}{35} (-1)^{-5} + C
   \]

   Since \( (-1)^{-5} = -1 \):

   \[
   -8 = -\frac{3}{35} (-1) + C
   \]

   \[
   -8 = \frac{3}{35} + C
   \]

   Solve for \( C \):

   \[
   C = -8 - \frac{3}{35}
   \]

   Convert \(-8\) to a fraction with a common denominator of 35:

   \[
   -8 = -\frac{280}{35}
   \]

   So:

   \[
   C = -\frac{280}{35} - \frac{3}{35} = -\frac{283}{35}
   \]

3. **Write the particular solution:**

   \[
   y = -\frac{3}{35} x^{-5} - \frac{283}{35}
   \]

   Simplify the final function if possible:

   \[
   y = -\frac{3}{35} x^{-5} - \frac{283}{35}
   \]

Therefore, the function \( y(x) \) that satisfies the given differential equation and initial condition is:

\[
y = -\frac{3}{35} x^{-5} - \frac{283}{35}
\]

Would you like more details on any of these steps or have any other questions? Here are some related questions you might be interested in:

1. How do you solve a first-order differential equation?
2. What are initial conditions in differential equations?
3. How do you integrate power functions?
4. What is the importance of integrating factors in differential equations?
5. How do you handle differential equations with multiple initial conditions?

**Tip:** Always check your work by substituting the initial condition back into the solution to ensure consistency.

Learn how to solve the differential equation dy/dx = (3/7)x^-6 with the initial condition y(-1) = -8. This problem involves differential equations and integration, focusing on the power rule for integration. Suitable for college-level students.

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