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.