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To determine which ODE corresponds to the given general solution, let's first analyze the general solution:

### Given General Solution:
\[
y = c_1 e^{2x} + c_2 e^{-5x}
\]

### Characteristic Equation:
For a second-order linear homogeneous differential equation of the form:
\[
y'' + by' + cy = 0
\]
the solution is based on the roots of the characteristic equation:
\[
r^2 + br + c = 0
\]

For the given general solution, the roots are \( r = 2 \) and \( r = -5 \).

So, the characteristic equation corresponding to these roots would be:
\[
(r - 2)(r + 5) = 0
\]
Expanding this:
\[
r^2 + 3r - 10 = 0
\]

This gives us the corresponding differential equation:
\[
y'' + 3y' - 10y = 0
\]

### Matching the Correct ODE:
Now, let's match this to the given options:

1. **\( y'' + 6y' + 10y = 0 \)**  
   - Characteristic equation: \( r^2 + 6r + 10 = 0 \).  
   - Roots: \( r = -3 \pm i \), not matching the given roots.

2. **\( y'' + 6y' + 12y = 0 \)**  
   - Characteristic equation: \( r^2 + 6r + 12 = 0 \).  
   - Roots: \( r = -3 \pm i\sqrt{3} \), not matching the given roots.

3. **\( y'' + 6y' + 15y = 0 \)**  
   - Characteristic equation: \( r^2 + 6r + 15 = 0 \).  
   - Roots: \( r = -3 \pm 2i \), not matching the given roots.

4. **\( y'' + 6y' + 8y = 0 \)**  
   - Characteristic equation: \( r^2 + 6r + 8 = 0 \).  
   - Roots: \( r = -2 \pm -4\), not matching the given roots.

### Conclusion:
The correct ODE from the provided options is:

Would you like further elaboration on how to solve these types of equations or other aspects of the problem?

### Relative Questions:
1. How do you find the characteristic equation of a differential equation?
2. What are the methods to solve second-order linear homogeneous differential equations?
3. How do initial conditions affect the solution of a second-order ODE?
4. Can complex roots occur in the characteristic equation? What would the general solution look like then?
5. What are the real-world applications of second-order differential equations?

### Tip:
Always check the roots of the characteristic equation to ensure they match the exponents in the general solution for homogeneous linear differential equations.

Learn how to identify the ordinary differential equation (ODE) corresponding to a given general solution. Understand the characteristic equation and its roots, which determine the structure of the ODE. Suitable for advanced high school and college students studying differential equations.

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