The problem you're working on involves solving a system of differential equations. You already selected the correct answer for $x(t)$, which is:

$x(t) = C_1 \cos(2t) + C_2 \sin(2t) + \frac{1}{5} e^{-t} + e^t$

Now, the next step asks to find $y(t)$ so that $y(t)$ and $x(t)$ together form a general solution to the system of differential equations.

Given that the original system is:

$2x' + y' = 7x + 6y + e^{-t}$ $x' + y' = -3x - 2y + e^t$

Step-by-Step Approach:

1. Express $y'(t)$: To solve for $y(t)$, we can substitute $x(t)$ into one of the original equations. Let's use the second equation: $x'(t) + y'(t) = -3x(t) - 2y(t) + e^t$ Solve for $y'(t)$: $y'(t) = -3x(t) - 2y(t) + e^t - x'(t)$

2. Substitute $x(t)$ and its derivative:

   • Derive $x(t)$ to find $x'(t)$.
   • Plug $x(t)$ and $x'(t)$ back into the equation to isolate $y'(t)$.
   • Then integrate to find $y(t)$.

Would you like me to continue with the detailed solution for $y(t)$?

Related Questions:

1. What is the derivative of $x(t)$ in this case?
2. How would you solve the system of equations if the coefficients were different?
3. Why is solving systems of differential equations important in real-world applications?
4. What methods are there for solving coupled differential equations, besides elimination?
5. Could we use matrix methods (like eigenvalue approaches) to solve this system?

Tip:

When solving systems of differential equations, always check if matrix methods like eigenvalue analysis can simplify the solution process, especially for linear systems.