 Show that 
y = 
6
7
ex + e−6x
 is a solution of the differential equation 
y' + 6y = 6ex.

We are given the differential equation:

\[
y' + 6y = 6e^x
\]

and the proposed solution:

\[
y = \frac{6}{7} \left( e^x + e^{-6x} \right)
\]

We need to verify that this expression for \( y \) is indeed a solution to the differential equation by finding \( y' \) (the derivative of \( y \)) and substituting both \( y \) and \( y' \) into the equation.

### Step 1: Find \( y' \)

Differentiate \( y = \frac{6}{7} \left( e^x + e^{-6x} \right) \) with respect to \( x \):

\[
y' = \frac{6}{7} \left( \frac{d}{dx}(e^x) + \frac{d}{dx}(e^{-6x}) \right)
\]

The derivatives of the exponential functions are:

\[
\frac{d}{dx}(e^x) = e^x \quad \text{and} \quad \frac{d}{dx}(e^{-6x}) = -6e^{-6x}
\]

Thus:

\[
y' = \frac{6}{7} \left( e^x - 6e^{-6x} \right)
\]

### Step 2: Substitute \( y \) and \( y' \) into the differential equation

The differential equation is:

\[
y' + 6y = 6e^x
\]

Substitute \( y' = \frac{6}{7} \left( e^x - 6e^{-6x} \right) \) and \( y = \frac{6}{7} \left( e^x + e^{-6x} \right) \) into this equation:

\[
\frac{6}{7} \left( e^x - 6e^{-6x} \right) + 6 \cdot \frac{6}{7} \left( e^x + e^{-6x} \right) = 6e^x
\]

Distribute the terms:

\[
\frac{6}{7} \left( e^x - 6e^{-6x} \right) + \frac{36}{7} \left( e^x + e^{-6x} \right)
\]

Combine like terms:

\[
\frac{6}{7} \left( e^x - 6e^{-6x} \right) + \frac{36}{7} \left( e^x + e^{-6x} \right) = \frac{6}{7} \left( e^x \right) - \frac{36}{7} \left( e^{-6x} \right) + \frac{36}{7} \left( e^x \right) + \frac{36}{7} \left( e^{-6x} \right)
\]

Simplify:

\[
\frac{6 + 36}{7} e^x + \frac{-36 + 36}{7} e^{-6x} = \frac{42}{7} e^x = 6e^x
\]

The terms involving \( e^{-6x} \) cancel out, and we are left with:

\[
6e^x = 6e^x
\]

This verifies that the given function \( y = \frac{6}{7} \left( e^x + e^{-6x} \right) \) is a solution to the differential equation.

### Conclusion:
Yes, \( y = \frac{6}{7} \left( e^x + e^{-6x} \right) \) is a solution of the differential equation \( y' + 6y = 6e^x \).

---

Do you want further details or have any questions?

### Related Questions:
1. How do you differentiate \( e^{ax} \) for any constant \( a \)?
2. What are the conditions for a function to be a solution of a linear differential equation?
3. Can we solve this differential equation using an integrating factor?
4. What is the general solution to a first-order linear differential equation?
5. How does the principle of superposition apply to linear differential equations?

### Tip:
When verifying solutions to differential equations, always remember to simplify carefully to ensure that terms cancel out correctly.

Show that y = (6/7)(e^x + e^(-6x)) is a solution of the differential equation y' + 6y = 6e^x.

Learn how to verify that y = (6/7)(e^x + e^(-6x)) is a solution to the first-order linear differential equation y' + 6y = 6e^x. The step-by-step solution covers differentiation of exponential functions and substitution into the equation. Suitable for students in Grades 11-12.

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