The given differential equation is:

$y' = y^7 - e^{4t} y^7, \quad y(0) = 1$

Step 1: Simplify the equation

Factor out $y^7$ on the right side:

$y' = y^7 \left(1 - e^{4t}\right).$

This is a separable differential equation.

Step 2: Rewrite in separable form

We can write:

$\frac{dy}{y^7} = \left(1 - e^{4t}\right) \, dt.$

Step 3: Integrate both sides

Left side:

$\int y^{-7} \, dy = \frac{y^{-6}}{-6} = -\frac{1}{6y^6}.$

Right side:

$\int \left(1 - e^{4t}\right) \, dt = t - \frac{e^{4t}}{4} + C,$

where $C$ is the constant of integration.

Step 4: Combine the results

We have:

$-\frac{1}{6y^6} = t - \frac{e^{4t}}{4} + C.$

Step 5: Solve for $y$

First, isolate $y^6$:

$\frac{1}{6y^6} = \frac{e^{4t}}{4} - t - C.$

Then:

$y^6 = \frac{6}{4e^{4t} - 24t - 24C}.$

Taking the sixth root on both sides:

$y(t) = \left(\frac{6}{4e^{4t} - 24t - 24C}\right)^{\frac{1}{6}}.$

Step 6: Apply the initial condition $y(0) = 1$

At $t = 0$, we know that:

$1 = \left(\frac{6}{4e^{0} - 24(0) - 24C}\right)^{\frac{1}{6}} = \left(\frac{6}{4 - 24C}\right)^{\frac{1}{6}}.$

Raising both sides to the 6th power:

$6 = 4 - 24C.$

Solve for $C$:

$24C = -2 \quad \Rightarrow \quad C = -\frac{1}{12}.$

Step 7: Final solution

Substitute $C = -\frac{1}{12}$ into the equation:

$y(t) = \left(\frac{6}{4e^{4t} - 24t + 2}\right)^{\frac{1}{6}}.$

This is the solution to the differential equation.

Do you have any questions or would you like further clarification?

Here are five related questions:

1. How do you determine if a differential equation is separable?
2. What are other methods to solve non-separable differential equations?
3. How does the initial condition affect the general solution?
4. What happens if the initial condition is not specified?
5. Can numerical methods be used to solve this problem?

Tip: When solving differential equations, always check if you can separate variables first—this often makes solving easier.