The given problem is asking to verify, by substitution, that the function $y = x^7$ is a solution of the differential equation:

$y' = 7x^6 \cdot y = x + 5$

Steps to Verify:

1. Given function: $y = x^7$

   First, compute the first derivative of $y$ with respect to $x$.

   $y' = \frac{d}{dx}(x^7) = 7x^6$

2. Substitute $y = x^7$ and $y' = 7x^6$ into the differential equation.

   The differential equation given is:

   $y' = 7x^6 \cdot y$

   Substituting $y = x^7$ into this equation:

   $y' = 7x^6 \cdot x^7 = 7x^{13}$

   However, on the right-hand side, the equation states:

   $y' = x + 5$

   Clearly, $7x^{13}$ does not equal $x + 5$. Therefore, the function $y = x^7$ does not satisfy the given differential equation.

Conclusion:

The function $y = x^7$ is not a solution to the differential equation $y' = 7x^6 \cdot y = x + 5$.

Would you like further details or have any questions on this process?

Here are 5 related questions to explore this topic:

1. How would you solve a first-order linear differential equation like this analytically?
2. What methods can be used to find particular solutions to non-homogeneous differential equations?
3. What role does substitution play in verifying solutions to differential equations?
4. How does one handle higher-order differential equations in a similar verification process?
5. What are common techniques to simplify complex differential equations?

Tip: When solving differential equations, always check if both sides of the equation match after substitution to verify a valid solution!