7. y = e ^ (- x) * (c_{1} * cos 3x + c_{2} * sin 3x)

The given equation is:

\[
y = e^{-x} \left( c_1 \cos(3x) + c_2 \sin(3x) \right)
\]

This looks like the general solution to a second-order linear differential equation with constant coefficients. It represents a combination of exponential decay and oscillatory behavior due to the trigonometric terms involving \( \cos(3x) \) and \( \sin(3x) \). Constants \( c_1 \) and \( c_2 \) depend on initial conditions.

### Analysis:

- The exponential factor \( e^{-x} \) causes the solution to decay as \( x \to \infty \).
- The trigonometric terms \( \cos(3x) \) and \( \sin(3x) \) suggest periodic oscillations with frequency 3.
- The constants \( c_1 \) and \( c_2 \) will be determined based on initial conditions provided by the problem.

This type of solution typically arises from solving a differential equation of the form:

\[
y'' + 2y' + 10y = 0
\]

Do you need help with the initial conditions or finding a more specific form of this solution?

### Related Questions:

1. What are the initial conditions required to determine \( c_1 \) and \( c_2 \)?
2. How does the frequency of oscillation (3) affect the solution?
3. What is the differential equation corresponding to this solution?
4. How does the exponential term \( e^{-x} \) influence the long-term behavior?
5. Can the solution be expressed in another equivalent form?

**Tip**: Solutions involving exponentials and trigonometric functions often appear in mechanical systems like damped oscillators.

Explore the solution to the second-order differential equation y = e^(-x) (c1 * cos(3x) + c2 * sin(3x)). This problem covers key concepts such as exponential decay, oscillatory behavior, and trigonometric functions, making it ideal for college-level math students.

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