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 $y' + 6y = 6e^x$.

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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.