We are given the differential equation:

$\frac{dy}{dx} = x^2 y^{-5}$

and the initial condition $y(0) = 8$. This is a separable differential equation, so we can solve it by separating the variables $x$ and $y$, and then integrating both sides.

Step 1: Separate Variables

We rearrange the equation so that all terms involving $y$ are on one side and all terms involving $x$ are on the other side:

$y^5 \, dy = x^2 \, dx$

Step 2: Integrate Both Sides

We now integrate both sides:

$\int y^5 \, dy = \int x^2 \, dx$

The integrals are straightforward:

$\frac{y^6}{6} = \frac{x^3}{3} + C$

where $C$ is the constant of integration.

Step 3: Solve for $y$

We multiply both sides by 6 to simplify:

$y^6 = 2x^3 + 6C$

Now, we solve for $y$:

$y = \left(2x^3 + 6C\right)^{1/6}$

Step 4: Apply the Initial Condition

To find $C$, we use the initial condition $y(0) = 8$:

$8 = \left(2(0)^3 + 6C\right)^{1/6}$

Simplifying:

$8 = (6C)^{1/6}$

Raise both sides to the 6th power:

$8^6 = 6C$

$262144 = 6C$

Solve for $C$:

$C = \frac{262144}{6} = 43690.67$

Final Solution

Substitute $C$ back into the expression for $y$:

$y = \left(2x^3 + 6 \times 43690.67\right)^{1/6}$

This is the solution to the differential equation.

Would you like more details on any specific step or have further questions?

---

Here are five related questions to deepen understanding:

1. How would the solution change if the initial condition were different?
2. What if the equation had an additional term, such as $\frac{dy}{dx} = x^2 y^{-5} + x$?
3. Can you explain the concept of separable differential equations in more detail?
4. What is the significance of the constant of integration in solving differential equations?
5. How would you check if the solution is valid for all values of $x$?

Tip: Always double-check your initial conditions when solving differential equations to ensure the constant of integration is correctly determined!