To determine which functions are also solutions to the system of differential equations given that $\mathbf{x}_1(t)$ and $\mathbf{x}_2(t)$ are solutions, let's analyze each option.

In a linear system of the form $\mathbf{x}' = A \mathbf{x}$, if $\mathbf{x}_1(t)$ and $\mathbf{x}_2(t)$ are solutions, any linear combination of these solutions (e.g., $c_1 \mathbf{x}_1(t) + c_2 \mathbf{x}_2(t)$ for constants $c_1$ and $c_2$) will also be a solution due to the superposition principle in linear systems. This principle holds because differential equations of this form are linear and homogeneous.

Analysis of Each Option:

1. $3 \mathbf{x}_1(t)$: This is a linear combination (a scalar multiple of $\mathbf{x}_1(t)$), so it must be a solution.
2. $\mathbf{x}_1(t) \cdot \mathbf{x}_2(t)$: This represents a pointwise product of the vectors, which is not a linear combination. Therefore, it is not necessarily a solution.
3. $-4.4 \mathbf{x}_2(t) + 5.2 \mathbf{x}_1(t)$: This is a linear combination of $\mathbf{x}_1(t)$ and $\mathbf{x}_2(t)$, so it must be a solution.
4. $2 \mathbf{x}_1(t) + 7 \mathbf{x}_2(t)$: This is also a linear combination, so it must be a solution.
5. $-7.6 \mathbf{x}_1(t) + 4.4 \mathbf{x}_2(t) + 7$: This includes a constant term (7), which is not a part of a homogeneous solution since we require the solutions to satisfy the differential equation without any constant terms.
6. $(5.4 \mathbf{x}_1(t) + 7.8 \mathbf{x}_2(t)) - 3.2 (\mathbf{x}_1(t) - \mathbf{x}_2(t))$: This can be expanded and simplified as a linear combination of $\mathbf{x}_1(t)$ and $\mathbf{x}_2(t)$, so it must be a solution.

Conclusion:

The correct answers are:

• $-4.4 \mathbf{x}_2(t) + 5.2 \mathbf{x}_1(t)$
• $2 \mathbf{x}_1(t) + 7 \mathbf{x}_2(t)$
• $(5.4 \mathbf{x}_1(t) + 7.8 \mathbf{x}_2(t)) - 3.2 (\mathbf{x}_1(t) - \mathbf{x}_2(t))$

These options satisfy the requirements of being linear combinations of the given solutions.

Would you like further details on this solution?

1. Why does the superposition principle apply to homogeneous linear systems?
2. How would adding a constant term affect the solution to a homogeneous differential equation?
3. Can non-linear combinations ever satisfy linear differential equations?
4. Why does the term $\mathbf{x}_1(t) \cdot \mathbf{x}_2(t)$ not work as a solution here?
5. What would change if the system were non-homogeneous?

Tip: When analyzing solutions to linear systems, remember that only linear combinations (without constants) of known solutions will generally form new solutions in homogeneous equations.