Use the elimination method to find a general solution for the given linear system, where differentiation is with respect to t. x'+6y=0 x'-y'=0

We are given the system of differential equations:

\[
x' + 6y = 0 \quad \text{(1)}
\]
\[
x' - y' = 0 \quad \text{(2)}
\]

Our goal is to use the elimination method to find a general solution.

### Step 1: Solve for \(x'\) from one of the equations
From equation (2), we have:
\[
x' = y' \quad \text{(3)}
\]

### Step 2: Substitute \(x'\) from equation (3) into equation (1)
Substitute \(x' = y'\) into equation (1):
\[
y' + 6y = 0
\]

This is a first-order linear differential equation for \(y\).

### Step 3: Solve the differential equation for \(y\)
The equation \(y' + 6y = 0\) is separable, and can be rewritten as:
\[
\frac{dy}{dt} = -6y
\]

Separating variables and integrating:
\[
\int \frac{1}{y} \, dy = \int -6 \, dt
\]

This gives:
\[
\ln |y| = -6t + C_1
\]

Exponentiating both sides:
\[
|y| = e^{-6t + C_1} = e^{C_1} e^{-6t}
\]

Let \( C_2 = e^{C_1} \), then:
\[
y(t) = C_2 e^{-6t}
\]

### Step 4: Solve for \(x\) using equation (3)
From equation (3), \(x' = y'\). We know that:
\[
y' = -6C_2 e^{-6t}
\]

Therefore, \(x' = -6C_2 e^{-6t}\). Now integrate this to find \(x\):
\[
x(t) = \int -6C_2 e^{-6t} \, dt = C_2 e^{-6t} + C_3
\]

### General Solution
Thus, the general solution for the system is:
\[
x(t) = C_2 e^{-6t} + C_3
\]
\[
y(t) = C_2 e^{-6t}
\]

Do you want further details or have any questions about these steps?

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Here are 5 related questions:
1. How can we verify that this solution satisfies both differential equations?
2. What is the significance of the constants \(C_2\) and \(C_3\) in the solution?
3. How would the system change if the coefficient of \(y\) in the first equation was different?
4. Can we solve this system using matrix methods instead of elimination?
5. What happens to the solution as \(t \to \infty\)?

**Tip**: When solving linear differential equations, always check if you can simplify the system using substitution or elimination methods to reduce the complexity.

Use the elimination method to find a general solution for the given linear system, where differentiation is with respect to t: x' + 6y = 0, x' - y' = 0.

Find the general solution for the linear system of differential equations x' + 6y = 0 and x' - y' = 0 using the elimination method. This problem covers the use of separable differential equations, leading to general solutions for x(t) and y(t). Suitable for undergraduate mathematics students.

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