The sum of powers of integers given by $\sum_{k=1}^x k^n$ is known as a power sum. We aim to find a differential equation such that this sum is a solution.

Step-by-Step Approach:

1. Expression for the Sum: The sum $S(x, n) = \sum_{k=1}^x k^n$ represents the sum of the first $x$ integers raised to the power $n$. This can be approximated by a polynomial in $x$.

   A known formula for this sum is: $S(x, n) = \sum_{k=1}^x k^n \approx \frac{x^{n+1}}{n+1} + O(x^n)$ This approximation is valid for large $x$, and it suggests the sum behaves roughly like a polynomial in $x$ with degree $n+1$.

2. Differentiation: We can try to differentiate the sum and see if we can formulate a differential equation that this sum satisfies. Differentiating $S(x, n)$ with respect to $x$ will yield: $\frac{d}{dx} \left( \sum_{k=1}^x k^n \right) = \frac{d}{dx} \left( \frac{x^{n+1}}{n+1} + O(x^n) \right) = x^n$

   Hence, we can conclude that: $\frac{d}{dx} S(x, n) = x^n$

3. Forming the Differential Equation: The equation we obtained is: $\frac{d}{dx} S(x, n) = x^n$ This means the sum $S(x, n)$ is the solution to the differential equation: $\frac{d}{dx} S(x, n) = x^n$

   To find the general solution, integrate $x^n$ with respect to $x$: $S(x, n) = \int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ where $C$ is a constant of integration. The constant can be determined based on boundary conditions, but for now, the solution to the differential equation is: $S(x, n) = \frac{x^{n+1}}{n+1} + C$

Thus, the sum $S(x, n) = \sum_{k=1}^x k^n$ satisfies the differential equation: $\frac{d}{dx} S(x, n) = x^n$

Would you like further details on the solution or the derivation process? Here are a few related questions to explore:

1. How does the constant $C$ change based on different boundary conditions?
2. What is the relationship between this sum and the Bernoulli polynomials?
3. How does the power sum behave for large values of $n$?
4. Can you derive a recursive formula for $\sum_{k=1}^x k^n$?
5. How can this approach be extended to sums involving other types of functions, such as $\sum_{k=1}^x k^n f(k)$?

Tip: To generalize sums of powers of integers, consider using generating functions or Bernoulli numbers to find more exact closed-form expressions.