The problem requires solving the third-order linear differential equation:

$\frac{d^3y(t)}{dt^3} + 2\frac{d^2y(t)}{dt^2} + 9\frac{dy(t)}{dt} + 2y(t) = f(t),$ where $f(t) = 3\sin(t) + 2\cos(3t)$.

We need to find $y(t)$, and the solution should include one sine term and one cosine term.

Solution Approach:

1. Solve the homogeneous equation: $\frac{d^3y(t)}{dt^3} + 2\frac{d^2y(t)}{dt^2} + 9\frac{dy(t)}{dt} + 2y(t) = 0.$ This involves finding the characteristic equation and its roots.

2. Find the particular solution: Use $f(t) = 3\sin(t) + 2\cos(3t)$, assuming a solution of the form: $y_p(t) = A\sin(t) + B\cos(t) + C\sin(3t) + D\cos(3t).$

3. Combine homogeneous and particular solutions: The total solution is: $y(t) = y_h(t) + y_p(t),$ and include only the necessary terms as specified in the question.

Let me calculate and finalize the solution. sympy import symbols, Function, Eq, sin, cos, dsolve, Derivative

Define the variables and function

t = symbols('t') y = Function('y')(t)

Define the given differential equation

f_t = 3sin(t) + 2cos(3t) # Forcing function eq = Eq(Derivative(y, t, 3) + 2Derivative(y, t, 2) + 9Derivative(y, t, 1) + 2y, f_t)

Solve the differential equation

solution = dsolve(eq, y)

solutionThe general solution for the given differential equation is:

$y(t) = y_h(t) + y_p(t),$ where:

1. $y_h(t)$ is the solution to the homogeneous equation. It is represented with arbitrary constants $C_1$, $C_2$, and $C_3$ in the exponential and trigonometric terms.
2. $y_p(t)$ is the particular solution. From the computation, it includes:

$y_p(t) = -\frac{3}{8}\cos(t) - \frac{1}{8}\cos(3t).$

For the specific question: Since the solution should include one sine and one cosine term, we focus on simplifying and isolating terms relevant to $f(t)$. Would you like to further isolate and analyze these terms or simplify them further?

Related Questions:

1. How do you derive the homogeneous solution from the characteristic equation?
2. What methods can be used to verify the solution of a differential equation?
3. What is the role of the forcing function $f(t)$ in determining the particular solution?
4. How does the method of undetermined coefficients apply to solving $y_p(t)$?
5. Can you explain how to interpret the constants $C_1, C_2, C_3$ in the solution?

Tip: When solving differential equations, always verify your solution by substituting it back into the original equation.