The exercise you're working on asks to determine the differential equation whose solution is given by the following equations:

1. $(y - 5)^2 = cx$
2. $y = c_1 + c_2x + c_3x^2 + c_4x^3$
3. $cx^2 + y^2 = 4$
4. $y = c_1e^{-x} + c_2e^{2x} + c_3e^x$

Let's break them down:

1. For $(y - 5)^2 = cx$:

   • Differentiate both sides with respect to $x$.

2. For $y = c_1 + c_2x + c_3x^2 + c_4x^3$:

   • Differentiate multiple times to eliminate constants $c_1, c_2, c_3, c_4$.

3. For $cx^2 + y^2 = 4$:

   • Differentiate both sides with respect to $x$.

4. For $y = c_1e^{-x} + c_2e^{2x} + c_3e^x$:

   • Differentiate multiple times and eliminate the constants $c_1, c_2, c_3$.

Would you like detailed solutions for each of these steps, or do you have a specific question regarding one of the problems?

Here are 5 additional questions for further exploration:

1. How do you eliminate constants in a general differential equation?
2. What techniques are commonly used for differentiating exponential functions?
3. How would you approach non-linear differential equations like problem 1?
4. What role do initial conditions play in determining arbitrary constants in solutions?
5. Can you explain the difference between general and particular solutions of differential equations?

Tip: When eliminating constants, remember that differentiating more than once is often necessary to fully remove all arbitrary constants.