What type of solutions does the following set of differential equations have?

dx/dt = -2x,
dy/dt = -8x-6y

(a) x and y both tend monotonically to zero
(b) x and y oscillate and the oscillations become larger with time
(c) x and y oscillate and the oscillations become smaller with time
(d) x and y both tend monotonically to infinity

To analyze the system of differential equations given by:

1. \( \frac{dx}{dt} = -2x \)
2. \( \frac{dy}{dt} = -8x - 6y \)

### Step 1: Solve for \( x(t) \)

The first equation can be solved directly:

\[
\frac{dx}{dt} = -2x
\]

This is a separable equation. Rearranging gives:

\[
\frac{1}{x}dx = -2dt
\]

Integrating both sides:

\[
\ln |x| = -2t + C \implies x(t) = C_1 e^{-2t}
\]

where \( C_1 = e^C \) is a constant.

### Step 2: Substitute \( x(t) \) into the equation for \( y(t) \)

Now, substitute \( x(t) \) into the second equation:

\[
\frac{dy}{dt} = -8x - 6y
\]

Substituting \( x(t) \):

\[
\frac{dy}{dt} = -8(C_1 e^{-2t}) - 6y
\]

This is a linear first-order differential equation in \( y \). To solve it, we can use an integrating factor. The equation can be rewritten as:

\[
\frac{dy}{dt} + 6y = -8C_1 e^{-2t}
\]

The integrating factor is:

\[
\mu(t) = e^{\int 6 dt} = e^{6t}
\]

Multiplying through by the integrating factor:

\[
e^{6t} \frac{dy}{dt} + 6e^{6t} y = -8C_1 e^{4t}
\]

The left-hand side is the derivative of \( e^{6t} y \):

\[
\frac{d}{dt}(e^{6t} y) = -8C_1 e^{4t}
\]

Integrating both sides gives:

\[
e^{6t} y = -2C_1 e^{4t} + C_2
\]

Thus,

\[
y(t) = -2C_1 e^{-2t} + C_2 e^{-6t}
\]

### Step 3: Analyze the solutions

1. **For \( x(t) \)**: As \( t \to \infty \), \( x(t) = C_1 e^{-2t} \to 0 \).
2. **For \( y(t) \)**: As \( t \to \infty \),
   - The term \(-2C_1 e^{-2t} \to 0\).
   - The term \(C_2 e^{-6t} \to 0\).

Thus, both \( x(t) \) and \( y(t) \) tend monotonically to zero.

### Conclusion

The correct answer is:

**(a) x and y both tend monotonically to zero.**

### Questions for Further Exploration:
1. What are the implications of the decay rates in each equation?
2. How would the system behave if the coefficients were positive?
3. Can you derive the solutions using a different method, such as eigenvalues?
4. What is the stability of the fixed points in this system?
5. How would the system change if we introduced a nonlinear term?

### Tip
When solving systems of differential equations, always look for separation of variables or integrating factors to simplify the process.

What type of solutions does the following set of differential equations have? dx/dt = -2x, dy/dt = -8x-6y

Explore the solutions of the differential equations dx/dt = -2x and dy/dt = -8x - 6y. Discover how both x and y tend monotonically to zero, demonstrating key concepts in differential equations and linear stability. Ideal for high school students in advanced mathematics.

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