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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?

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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!

Verify by substitution that the given function y = x^7 is a solution of the given differential equation y' = 7x^6 * y = x + 5.

Verify by substitution that the function y = x^7 is a solution of the differential equation y' = 7x^6 * y = x + 5. Follow the step-by-step process of differentiation and substitution to check if the equation holds true.

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